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THE COVER nIDEX OF A GRAPH

by

Ram Prakash Gupta

University of North Carolina

Institute of Statistics Mimeo Series No. 562

December 1967

This research was supported by the Air ForceOffice of Scientific Research Contract Bo.AFOSR-68-1415 and also partially supportedby the National Science Foundation Grant No.GP-5790.

DEPARTMENT OF STATISTICS

University of North Carolina

Chapel Hill, N. C.

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1. Introduction. Let G be a graph (undirected, without loops) with

set of vertices V(G) and set of edges E(G). Any subset C of E(G) is

said to be a cover of G if each vertex YcV(G) is an endvertex of at

least one edge in C. The cover index of a graph G, denoted by k(G),

is defined to be the maximum number k such that there exists a de-

composition of E(G) into k mutually disjoint sets each of which is a

cover of G. The lower degree d(G) of G is the greatest lower bound

of the degrees of its vertices. (For the definitions of the cover

index k(D) and the lower degree d(D) of a directed graph D, see Section

2.) A graph G is said to be of multiplicity s or simply an s-graph is

no two of its vertices are joined by more than sedges.

In the present paper, we deal with the problem of determining

the cover index of a graph in terms of its lower degree and multipli­

city. For any graph G, we must have k(G) ~ d(G). Also, k(G) > 1 if

and only if d(G) ~ 1. O. Ore [9] proved that "for a graph G, k(G) > 2

if and only if d(G) ~ 2 and no connected component of G is a cycle

with odd number of edges". He then proposed [10] to establish an

analogue of this result for directed graphs. In Section 3, the follow­

ing general theorem is proved. Theorem 3.1 "If G is a locally finite

bipartite graph thenk(G) = d(G)". As a corollary to Theorem 3.1, we

then obtain Theorem 3.2 'Tor any locally finite directed graph D,

k(D) = d(D)". Evidently, Theorem 3.2 solves a more general problem

than that suggested by O. Ore. Further, we observe that Theorem 3.2,

and hence, a priori Theorem 3.1, implies a well-known theorem of J.

Petersen [11] and D. Konig [8].

In Section 4, considering the problem for arbitrary graphs (not

+ s".

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necess~rily bipartite) the following theorem is proved. Theoram 4.1

"If G is any s-graph which is locally finite, then d(G) 2: keG) 2: d(G)-s.

The bounds are best possible in the following cases: s > 1 and

- [r-l]d(G) = 2ms-r where r > 0, m > -- 2

In the last section, some remarks on the dU8,lity between the

concept of the cover index of a graph, introduced here, and the well-

known concept of the chromatic index of a graph are added.

2. Definitions and Notations; colorings; The Conc~pt of Alternating

Chains.

The method used in proving our results in this paper is based

on the concept of alternating chains introduced bJ' Petersen (1891) in

his invertigations on the existence of subgraphs '\Irhich has been an

effective tool for many problems in graph theory. In this section,

we first explain the basic concepts and notations which are used

throughout this paper, and then define the concept of colorings and

alternating chains.

Definitions and Notations: An undirected ~'aph or simply a graph

G is defined by a nonempty set V(G) of vertices aIld a set E(G) of edges

togetherwith a relationship which identifies each edge eeE(G) with an

unordered pair (v,u) of distinct vertices v,u€V(G), called its~

vertices, which the edge is said to join. In the following, the letter

G denotes, without any further specifications, a graph. (It should

be noted that by definition, we exclude from our c:onsideration graphs

which have 'loops I, 1. e., edges with coincident elldvertices.) Two

vertices are adjacent if they are joined by an edge; two edges are

adjacent if they have a common endvertex. A vertex v and an edge e are

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G is infinite.

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Two graphs are said to be edge-disjoint if they have no edge in

•

e. € E(G), and (3) e. joins v. 1 and vl.

l l l-vi E V(G), (2)

As a notational convention, if e is an edge joining the vertices

partial graph r:£ asubgraph of G is c?"lled a partial subgrp"ph of G. (It

Any subset E' of E(G) defines a partial graph G' of G such that

A graph G is finite if both V(G) and E(G) axe finite; otherwise

s ~ 0, then G is said to be of multiplicity s or simply an s-graph.

If no two vertices of a graph G 8xe joined by more than sedges,

we may have v = v' or u = u' or both v = v' and u = u'.

e' = (v',u') are two distinct edges, then, unless specifically stated,

and u then we write e = (v,u). We note that if e = (v,u) and

incident with e~'l.ch other if v is an endvertex of e...---.

A chain in a graph G is a finite sequence (we shall have no

is an edge of Gt if and only if both of endvertices are in Vt• A

V(G') = V(G) and E(G') = E'. Any (nonempty) subset V' of V(G) defines

a subgraph G' of G such that V(G t) = V', E(G') s E(G) and an edge of G

is implied that every edge of a partial subgraph Gt of G has the same

endvertices in G' as in G.)

where (1)

occasion to consider infinite chains) of the form

for each i=1,2, •.• ,r. (If all the vertices and edges of a graph G can

common. A nonempty family {Go' Gl, ... ,Gk_l } of mutually edge-disjoint

partial.graphs of a graph G is said to form a decomposition of G ifk.,.l

E(G) =.~ E(G.); we then writel-v l

(2.1) G = GO + Gl + ••• + ~-l

(2.2)

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be arranged in a chain then G is itself called a chain.) The chain

~(vo,vr) is said to connect its 'terminal vertices:' Vo and vr ' A

graph is connected if every two of its vertices al'e connected by a

chain. The relation of being connected in G is o'bviously an equivalence

relation. It, therefore, 'partitions' G into a f13LlJlily (G ) of graphsa

each of which is a maximal connected subgraph of 0 and is called a

connected component of G. A chain (2.2) is called a cycle if (1) it

has at least one edge, (2) its edges el ,e2" .. ,er are all distinct, and

(3) its last vertex vr coincides with its first VE!rtex YO'

In a graph G, the degree d(G,v) of a vertex v is the number of

edges that are incident with v, The lower degree of G, denoted by

d(G), is the greatest lower bound of the degrees of its vertices. Thus,

the lower degree of a graph is the largest number k such that there

are at least k edges incident with each of its vertices.

A gra.ph is said to be locally finite if the degree d{G.,v) of

each vertex v € V(G) is finite, Trivially, if G is locally finite,

then its lower degree d{Q) is finite.

Any set of edges C of a graph G, C~ E(G), is called a cover or

an edge-cover (for the vertices) of G if every vertex v € V{G) is

incident with at least one edge in C. Obviously, if G ha.s an 'isolated

vertex' (a vertex is isolated if its degree is zero) or if d{G) = 0,

then there can exist no cover of G.

The cover index of a graph G, denoted by keG), is defined as

. -follows: keG) = 0 if d{G) = 0; otherwise, keG) is the maximum number

k such that there exists a partition of E(G) into k sets.

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where each of the sets E.(i=O,l, ••• ,k-l) is a cover of G.1

A partial graph G' of G is called a covering graph of G if

d(G') 21, or, in other words, if its defining set of edges E(G') is

a cover of G. The cover index k(G) of a graph G may equivalently be

defined to be the maximum number k such that there exists a decomposition

(2.1) of G where each of the graphs G.(i=O,l, ... ,k-l) is a covering1

graph of G. Evidently, if d(G) = OJ then no such decomposition of G

exists and in that case we have, by definition, k(G) = O. Henceforth,

we may disregard the trivial case d(G) = 0 and assume tacitly, if

necessary, that d(G) 2 1.

A directed graph or briefly digraEh D is defined by a nonempty

set V(D) of vertices and a set A(D) of~ together with a relation­

ship Which identifies each arc with an ordered pair of (not necessarily

distinct) vertices which the arc is said to join. For each arc (y, u),

v is called its initial vertex ani u its terminal vertex; both v and

u being called its endvertices.

A digraph D is finite if both V(D) and A(D) are finite; otherwise,

D is infinite.

In a digraph D, the outward degree d+(D,v} of a vertex v is the

number of arcs in A(D) with initial vertex v and the inward degree

d-(D,v) of v is the number of arcs in A(D) with terminal vertex v. A

digraph D is locally finite if for each vertex v ~ V(D), the sum

d+(D,v) + d-(D,v) is finite.

The lower degree of D, denoted by d(D), is the largest number k

such that d+(D,v) 2 k, d-(D,v) ~ k for every vertex v € V(D). Obviously,

if D is locally finite, then its lower degree d(D) is finite.

For a digraph D, the notions of partial graph and subgraph are

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defined analogously as for undirected graphs. ~1US, a partial graph

of D is a digraph D' such that V(D') ::: V(D) and A(D') c:=. A(D). Two

digraphs are arc-disjoint if they have no arc in common. A nonempty

family (Do,Dl, •.. ,Dk

_l ) of mutually arc-disjoint partial graphs of a

digraph D is said to form a decomposition of D i:f' A(D) :::\jlA(Di ),l=O

and we write D ::: DO + Dl + ••• + Dk_l •

A set of arcs C of a digraph D, C c:=. A(D), :ls called a cover of

D if for each vertex v «V(D), there is in C at least one arc with

initial vertex v and at least one arc with terminal vertex v. A

partial graph D' of D is defined as suggested by O. Ore [10], to be a

covering graph of D if d(D') ~ 1 or, equivalent~r, if its defining set

of arcs A(D') is a cover of D.

The cover index of a digraph D, denoted by k(D), is defined

as follows: k(D) ::: 0 if d(D) ::: 0; otherwise, k(D) is the maximum

number k such that there exists a decomposition of D into k partial

graphs each of which is a covering graph of D; or equivalently, k(D)

is the maximum number k such that there exists a partition of A(D)

into k sets each of which is a cover of D. In the following, we may

disregard the case d(D) ::: 0 and assume that d(D) ~ 1.

p~

Colorings: Our objects in this ~aJlter necessitate the con-

sideration of partitions of the set of edges of ll. given graph. Con-

sider, therefore, for an undirected graph G, an ~l.rbitrary decomposition

of its set of edges E(G):

(2.4) EO' El , . - -, ~-l; E(G)::: UE i , E:LnEj ::: t.

It is convenient for us to think of each SE!t E. as representingl

A. color which we denote by the integer i. FurthE!r, we say that each

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edge e € E. is colored with the color i thus obtaining ~ coloring ofl

the edges of G by means of k distinct colors. Formally, any function

f which assigns to each edge e € E(G) an integer f(e)(O,l, ••. ,k-l)

is cFl.lled a coloring or more specifically, a k-coloring (of the edges)

of G. The integers O,l, •.. ,k-l are themselves referred to as colors

and if f(e) = i we say that the edge e is colored with the color i or

briefly that e is an i-edge. It is readily seen that there is a

natural one-to-one correspondence between k-colorings of G and de-

compositions of E(G) into k sets.

Any k-coloring of G is called admissible if every vertex v E V(G)

is an endvertex of at least one i-edge for each i = O,l, .•• ,k-l. It

is immediately seen that in a decomposition (2.4) of E(G), each E.l

(i=O,l, •.. ,k-l) is a cover of G if and only if its corresponding

k-coloring of G is admissible. Thus, the cover index of a graph G is

the maximum number k for which an admissible k-coloring of G exists.

Let f be any k-coloring of G. For any vertex v € V(G), we

denote by cf(v) or simply c(v), if no confussion is possible, the set

of colors which are assigned by f to the edges incident with v.

Clearly, cf(v)S (O,l, .•• ,k-l). The difference

(2.5) Ar(v) = (k - Icf(v) Dis called the deficiency of f at the vertex v. (lsi denotes the number

of elements in the set S.) For any subset V' of V(G), the sum

(2.6) Ar(V')= \' (k-lcf (v)1)v'rt'

is called the deficiency of f at V'. (By definition, ~f(V') = °if V' = t). The total deficiency of f, denoted by Af(G), is obtained

when the sum (2.6) is eXtended over all vertices of G, 1. e., when

V' = V(G). It is clear that a k-coloring f of a graph G is admissible

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if and only if its total deficiency Af(G) = 0. Thus, in order to prove

that k(G) ~ k, where k is some positive integer it is sufficient to

shOv1 that G possesses a k-coloring f whose total deficiency is zero.

The Concept of AIternating Chains: Any cha.in considered in

the following is assumed to have at least one edge and all of its

terms are to be distinct except that its last vertex may coincide with

one of the vertices preceding it.

Consider an arbitrary k-coloring (k ~ 2) of a graph G And let i

and j be any two fixed colors. A chain

(2.7) ~(vO,vr') = (vO,el,vl,e2, ••• ,vr_l,er,vr)' r ~ 1,

in G is called an (i,j)-alternating chain if its edges el ,e2, ••• ,er

are alternately colored with i and j, the first one being an i-edge.

An (i, j)-alternating chain (2.7) is called :suitable if it satis-

fies anyone of the following conditions:

(i) v = v .° r'(ii) e is an i-edge (respectively, j-edge) and there is no j-edge

r

(respectively, i-edge) incident vdth v ;r

(iii) e is an i-edge (respectively, j-edge) and there is anotherr

i-edge (respectively, j-edge) incident with v •r

A suitable (i,j)-alternating chain (2.7) is called minimal

if I1(VO'Vt ) = (vO,el'vl , ... ,et,vt ) is not suitable for any t,

1 < t < r-l. It is easy to observe that if G is finite, then, given

any i-edge el = (vO,vl)e E(G), a (minimal) suitable (i,j)-alternating

chain (2.7) can always be determined. (In fact, this can be done by

follovTing along an (i, j ) -alternating chain starting with el till one

of the conditions (i), (ii) or (iii) is satisfied.)

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The method of alternating chains, as used in this paper, consists

in the following: Suppose G is a finite graph and we want to show that

G possesses a k-coloring (k ~ 2) which is admissible. We start with

an arbitrary k-coloring f of G. If Ar(G) = 0, we a.re through. Other­

wise, choosing a vertex Vo E V(G) with Af(VO) > 0 and colors i and j

properly, a suitable (i,j)-alternating chain starting with the vertex

v0 is found and the colors i and j interchanged on all edges belonging

to this chain. Such a step may have to be taken more than once till

we obtain a k-coloring g (say) such that Ag(G) < Af(G). The assertion

is then proved. by a proper use of finite induction.

3. The Cover Index of a Bipartite Graph

A granh G is said to be bipartite (simple, C.Birge [I) if it

contains no cycle with odd number of edges. Alternatively, G is

bipartite if (and only if) its set of vertices V(G) can be decomposed

into two disjoint sets VI and Vft such that each edge e=(v t, VII) e: E(G)

joins a vertex v I E VI with a vertex VII E VII. These special kind of

graphs are of particular significance in graph theory and play a con­

siderable role in various applications. For instance, any (0,1)-

matrix or a family of subsets of a given set can be represented by a

bipartite gra,ph and graph theory becomes a handy tool for sever1'l,l

combinational problems relating to such systems. Also, we observe

that to any directed graph D, there corresponds, within isomorphism,

a unique bipartite graph GD (an arbitrary bipartite graph may not

correspond to any directed graph) defined as follows: to each vertex

v E V(D) there correspond two vertices v t,v- ftE V(GD) and to each arc

(v,u) e:A(D) there corresponds an edge (vt,u ll) eE(GD). Thus, any

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problem for directed grapm may be formulated asa problem for bipartite

graphs.

Let G be any graph. Cle?"rly, if d(G) = 0 or 1, then keG) = 0 or

1 respectively. O.Ore (1962) proved that "for ~~y graph G, keG) ~ 2 if

and only if d(G) ~ 2 and no connected component of G is ~. cycle with

odd number of edges." He then suggested [10] to establish an analogue

of this theorem for directed graphs. The solution to this problem

is provided by the following theorem "for any directed graph D,

keD) > 2 if and only if den) > 2", which we can l~OW prove. Indeed,- -this theorem is most conveniently derived as a corollary to the above

theorem of O. Ore. (The det~.ils of the proof wh:ich are simple, are

omitted.)

We consider below the general problem of d1etermining the cover

index of a bipartite graph which as is easily shl:>wn includes the problem

of determining the cover index of a directed gr~ph. We shall prove

now the following theorem which may be called a "Decomposition Theorem

for Bipartite Graphs."

Theorem 3.1 Let G be a b.ipartite graph which is locally finite

and let d(G) = k. Then, there exists a decompos:ition

(3.1) G = GO + Gl + ••• + Gk _l

such that each of the graphs G. is a covering grl~ph of G,Le.,~

~(Gi) 2 1 (i=O,l, .•• ,k-l). In other words, if G is a locally finite

bipartite grap~ then keG) = d(G).

Proof: If d(G) = 0 then the theorem is trivially true. Let,

therefore, d(G) = k > 1. Evidently, we must havl:! keG) :s k. Hence,

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to prove the theorem, we have to show that k(G) 2: }t or equivalently

that there exists a k-coloring f of G which is admissible, i.e.,

6r(G) = O. We shall infact prove the following lennna which is apparent­

ly stronger than the above statement.

Lemma 3.1 Let G be a locally finite bipartite graph and let

V' be any subset of V(G) such that d(G,v) 2: k, k 2: 1, for all v E V' .

Then, there exists a k-coloring f of G such that Af(V') = O.

Proof: The lemma will be proved first for the case in which G

is finite; then, assuming the lennna to be true for all finite graphs,

it will be proved, by using a well-known argument due to D. Konig

and S. Valko [7], when G is infinite (but locally finite).

G is finite: If k = 1, the lemma holds obviously. Suppose,

therefore, that k > 2. Consider k-colorings of the graph G. Since G~

finite, the number of distint k-colorings of G is finite. Hence, there

must be a k-coloring f of G such that Af(V') is minimal, i.e.,

A_(V') < 6. (V') for any other k-coloring g of G. We shall show that~. - g

6r(V') = O.

Suppose, if possible, that 6.f (V') > O. Then, there will be a

vertex V o E V' such that Ar(v0) > 0 "Which is equivalent to saying that

there is a color j such that j fi cf(vO). Since, now, at most k-l

colors are assigned to the edges incident with Vo and by assumption

d(G,vo) 2: k, there must be a color i such that there are (at least)

two i-edges incident with vO' Now, we determine a suitable (i,j)­

alternating chain ~(vO,vr) starting with the vertex vo. Let g denote

the k-coloring of G obtained from f by interchanging the colors i and

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G is infinite: Let us first assume that Gf is connected. Since,

assumption that G is ~, bipartite graph. Hence, It is seen that, since

the lemma holds for finite graphs, it follows tt.~t G has k-coloringsn

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Let any such k-coloring f of G be calledIlL n

Now, if n < m, G is a subgraph of G and each proper k-n III

'proper' •

f n such that Ar (V~) = 0, but the number of such k-colorings of Gnnis obviously finite.

coloring f of G implies a proper k-coloring f of G which is them m n n

and we write f < f. Now, the proper k-colorizllgs of the graphsn m

restriction of the coloring of G . f is called an extension of fm' m n

G2

, G3"" imply proper k-colorings of Gl and si.nce Gl has only a

moreover, G is locally finite, the set of vertices V(G) of G is

enumerable. Let vl'v2 ,v3'.'.' be an enumeration of V(G) in some

order. Let Vn (n=1,2, ... ) denote the set of all vertices v such

that either v « (v l'v2' •• • ,vn) or v is adjacent to some vertex1rt;.~

vi' i=1,2, ••• , n. Denote by Gn the sUbgr~ of Gf defined by the set

of vertices Vn and let V~ =V'n(vl ,v2 , ... ,vn ). Now, it is evident

that the graphs G are finite and d(G ,v) > k for each v E V'. Sincen n - n

j on all edges belonging to the chain ~(vO,vr)' Since the chain is

suitable, it is easily seen that Icg (v)l2: Icf(v)1 for all vertices

except possibly when v = v. We now observe that v cannot coincideo r

with YO' In fact, if vr = V o then since j ~ cf(vO), ~(vO,vr) would

be a cycle in G with odd number of edges which contradicts the

there were two i-edges incident with V o with re~;pect to f, cg(vo) =

cf(vO) U (j) ~ cf(vO). Hence, by definition (2.6), it follows that

Ag (V') < Ar (V') which is contradictory to the choice of f. Hence,

we must have Ar(V') = O. This proves the lemma for finite graphs.

We define a k-coloring f of G as follows: for

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finite number of proper k-colorings, there must be one among them

a decomposition of G into k covering graphs. We start with an arbi-

It is clear that Af(V')=O.if e E E(G ).a

The theorem now follows immediately if we take V' = V(G) in

fix one such coloring f l of Gl • We consider now only the extensions

of this fixed coloring. Each of these implies a proper k-coloring

which is restriction of infinitely many proper k-colorings. Let us

that Af (V~) = O.a

every e E E(G), f(e) = f (e)a

have just proved, each of the graphs G has a k-coloring f sucha a

of G2 , Hence, by the same reasoning as above, G2 must hl:'l,Ve a proper

k-coloring f 2 which is restriction of an infinity of proper k-coloring.

We proceed in this manner to obtain a sequence of k-colorings

f l ,f2, .•• such that fn

(n=1,2, ..• ) is a proper k-coloring of Gn and

f l < f 2 < f3

< ..• Now, we define a k-coloring f of G as follows:

for each e € E(G), f(e) = f (e) if n is the first index such thatn

e ~ E(Gn). Evidently, f is well-defined and we have Af(V ' ) = O.

If G is not connected, let (G ) be the family of connecteda

components of G. Let V' = v'nv(G ) for each index a. Then,a a

d(G ,v) > k for each v c VI and since G is connected, by what wea - a a

the above lemma. Hence, the proof of the theorem is also complete.

This completes the proof of the lemma.

trary k-coloring f l of G. If f l is not admissible then a suitable

alternating chain is found as indicated in the proof of the lemma.

actual determination of an admissible k-coloring of G or, equivalently,

Remark: Let G be a finite graph Which is bipartite and let

d(G) = k. The proof of Lemma 3.1 suggests an algorithm for /'In

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By interchanging the colors of all edges belonging to this chain,

we obtain a new k-coloring f 2 of G such that th.e total deficiency of

f 2 is less than the total deficiency of f l by at least 1... The process

is repeated with the new coloring (each time) till an admissible k-

coloring of G is obtained. Since G is finite, an admissible k-

coloring of G is obtained after a finite number of repititions of

the process. (It may also be noted that the above algorithm is quite

efficient and can be programmed convenie ntly for us e on electronic

computers.)

There are several important consequences of Theorem 3.1 which

we shall state below.

We first obtain the fo1.1.owing theorem which may be called a

"Decomposition Theorem for Directed Graphs".

Theorem 3.2 Let D be a directed graph which is locally finite

and let d(D) = k. Then, there exists a decomposition

(3.2) D = DO + Dl + ••• + Dk_l

such that each of the digraphs D. is a covering graph of D,i.e.,J.

d(D.) > 1 (i=O,l, ••. ,k-l). In other words, if D is a locally finiteJ. -

digraph then k(D) = d(D).

Proof: Consider the bipartite graph GD corresponding to the

given graph D as defined above. Clearly, GD is locally finite and

d(GD) = d(D). AlSO, it is seen that a set of arcs in D is a cover

of D if and only if its corresponding set of edges in GD is a cover

of GD so that we have evidently, k(GD) = k(D). Now, by Theorem 3.1,

we have k(GD) = d(GD) whence the theorem follows immediately.

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We say that a digraph D is regular of degree k if....l tl+ -

d' (D,v) = d (D, v) = k for each vertex v c V(D). Any partial graph-.--

D' of D is c?lled an r-factor of D if D' is regular of degree r. If,

in Theorem 3.2, we take D to be regular of degree k, then evidently,

each of the digraphs Di(i=O,l, ••• ,k-l) in the decomposition (3.2)

will be a l-factor of D. Hence, as a particular case of Theorem 3.2

we obt::dn the following:

Corollary 3.1 Let D be a directed graph which is regular of

degree k. Then, there exists a decomposition (3.2) of D such that

each of the digraphs D. (i=O,l, ••• ,k-l) is a l-factor of D.~

Consider an undirected graph G. The graph G is said to be

regular (homogeneous)C.Berge [lJ) of degree k if d(G,v) = k for each

vertex v E V(G). A partial gr?ph G' of G is called an r-factor of

G if G' is regular of degree r. It is known that if G is a regular

graph of (even) degree 2 k, then the edges of G can be so 'directed'

('oriented'), each edge being assigned a unique direction, that the

resulting digraph Gd, say, is regular of degree k. Clearl.v, jf a

partial graph Gci of Gd is an r-factor of Ga, then its corresponding

partial graph G' of G is a 2r-factor of G. Hence, from Corollary

3.1, we obtain further the following 'Tactorization Theorem," due

to Petersen [11].

Corollary 3.2 (Theorem of Petersen): Let G be a regular graph

of degree 2k. Then, there exists a decomposition (3.1) of G such

that each of the graphs Gi

(i=O,l, ••. ,k-l) is a 2-factor of G.

It may also be noted that the theorem of Petersen implies in

its turn Corollary 3.1.

16

The following result for regular bigartite graphs due to Konig

[8], which is I'! special case of a more general theorem of Konig and

P. Hall [6J, cl'ln also be shown to be equivalent to Corollary 3.1 or

the theorem of Petersen and hence, is a consequence of Theorem 3.2.

Corollary 3.3 (Theorem of Konig): Let G be a bipartite graph

which is regular of degree k. Then, there exists a decomposition

(3.1) of G such that each of the graphs G.(i=O,l, ••. ,k-l) is 2l-factor~

of G.

We observe that Theorem 3.1 is apparently stronger than Theorem

3.2 which implies the theorems of Petersen and Konig. It is not known

if the converse inplication also holds. Specifically, we may ask the

following question: "Is it true that the theorem of Petersen (or Konig)

implies Theorem 3.11" (The question was first put to the author by

Professor C. Berge in personal discussion.)

We state below another problem which seems to be of considerable

interest. A digraph D is said to be pseudosymmetr!.£ if i(D,v)=d-(D,v)

for each vertex v • V(D). We believe that the following statement

is true.

Conjecture: Let D be a locp,lly finite directed graph which is

pseu.dosynunetric and let d(D) = k. Then, there exi:sts a decomposition

of D into k covering graphs each of which is also JPseu.dosynunetric.

4. The Cover Index of an s -graph

The cover index k(G) of a graph G is the maximum number k such

that there exist k mutually disjoint covers of G. If the degree of a

vertex v € V(G) is k, then clearly, since any cover of G must contain

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at leFt,St one edge incident with v, the graph G cannot have more than k

mutually disjoint covers. Hence, it is obvious that keG) ~ d(G) where

d(G) is the greatest lower bound of the degrees of the vertices of G,

In the previous section, it was proved that if G is a locally finite

bipartite graph then the equality keG) = d(G) holds. However, we may

have in general keG) < d(G), as is shown by the following examples.

r-l]For any integers s ~ 1 and k = 2ms-r where r ~ 0, m > [2 + s,

we give examples of s-graphs G such that d(G) = k and keG) ~ d(G)-s.

To this end, consider a graph K with 2m+l vertices vO,vl , ••• ,v2m in

which each pair of distinct vertices are joined by exactly sedges.

Clearly, K is an s-graph which is regular of degree 2ms. If k=2ms-r

is even, then, by the theorem of Petersen (Corollary 3.2), K has a

partial graph G which is regular of degree k. Obviously, G is an s­

graph and d(G) = k. If keG) > k-s+l, then, since any cover of G must

contain at least m+l edges, we have IE(G) I = ik(2m+l) ~ (m+l)(k-s+l) or,

after simplification, m ~ [r~l] + s which contradicts the assumption

that m > [r;l] + s. Hence, keG) ~ d(G)-s. AlSO, if k = 2ms-r is odd,

then, it can be easily shown that K has a partial graph G such that

d(G, vO) = k+l and d(G,vi ) = k for i=1,2, ••• ,2m. Obviously, G is an

s-graph and d(G) = k. As above, it is seen that keG) ~ d(G) - s.

The main result proved below (Theorem 4.1) is that for any locally

finite s-graph G, keG) ~ d(G) - s. Evidently, in view of the above

examples, this bound cannot be improved in general.

Before proving Theorem 4.1, we first explain some notations and

terminological conventions. Consider a graph G, and let f be any color-

ing of G. As earlier, for any vertex v c V(G), cf(v) denotes the set

of colors i such that there is at least one i-edge incident with v.

18

Further, c~(v) or simply cr(v), r 2: 2, denotes the set of colors i such

that there are at least r i-edges incident with v, and cf(v) or simply

c(v) denotes the set of colors which do not belong to c(v).

In the following, the letters g,f,fO,fl , etc. denote colorings

of G. Let f be ar..y coloring of G and let IJ.( v, u) be an (i, j) -alternating

chain in G. Let g be defined as follows: for an~r edge e € E(G),

gee) = j if e belongs to lJ.(v,u) and f(e) = i,

= i if e belongs to J1(v,u) and fee) = j,

= fee) if e does not belong to Jl(v,u).

Then, we say that g is obtained from f by interchanging the colors

on lJ.(v,u).

For the sake of convenience, all suitable a~ternating chains

considered below are assumed to be minimal. Any minimal, suitable

(i,j)-alternating chain is referred to, for brevity, simply as an

(i,j)-chain. (As observed earlier, if there is an i-edge incident

with a vertex v, then an (i,j)-chain with v as its initial vertex

can always be determined.)

We are now prepared to prove the following:

Theorem 4.1: If G is an s-graph which is locally finite, then

d(G) 2: keG) 2: d(G) -s. The bounds are best possible in the following

cases: s 2: 1, and d(G) = 2ms - r where r > 0, m > [r~l] + s.

Proof: Let G be a.ny locally finite a-graph, and let d(G) = k

To complete the proof of the theorem, we have to prove that

keG) > d(G) - s, or equivalently, that G possesses a (k-s)-coloring

f which is admissible, the rest having been proved above. We sha~~

prove it only for the case when G is finite. The proof can then be

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19

extended to all infinite, locally finite graphs as we did in the case

of Lemma 3.1. Let, therefore, G be finite. If k - s .:s 1, then the

assertion holds obviously. Hence, we may assume that k - s 2: 2. We

start with an arbitrary (k-s)-coloring f of G. If f is not admissible,

Le., if t\-(G) > 0, then we shall show that by suitably IOOdifying the

coloring f, we can obtain another (k-s)-coloring g of G such that

6 g (G) < ~(G). Since G is finite, the assertion will then follow by

finite induction. The operations used in modifying the coloring f

are as follows:

(I) Interchange any two colors on all edges of G. (This is

equiValent merely to renumbering the colors.)

(II) Interchange colors on an (i,j)-chain t&(v,u) if j E c(v),L'

and/or there is an i-edge incident with y which does not belong to

~(v,u).

It is clear that if g is obtained from f by using (I) or (II)

any number of times, then A (G) < A_(G). M:>reover, we ha,ve evidently,g --y

the following:

Lemma 4.1: Let f be any coloring of G, and let l£(v,U) be an (i,j)-

chain in G. Let g be obtained from f by interchanging colors on p(v, u) .

Then, t1g (G) < ~(G) if j • Cf(v) and there is an i-edge incident with LJ-

,.'which does not belong to ~(v,u).

We now proceed with the proof of the inequality k(G) 2: d(G) - s.

Let f be any (k-s)-coloring of G which is not admissible. Then, there

is a vertex Vo E V(G) such that ~(vo) > 0 or equivalently, c(vo) f t.

Suppose that we also have c3(vo) f$. Let i E c3(vo), and let j E: C(Vo)

Consider an (i,j)-chain lJ.(vO'u) and let g be obtained from f by

interchanging the colors on lJ.(vO'u). Since

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i l € c (vO)' and for each t, 2 :s t :s m, there is a (unique) iniex(ii)

(iii)

ing conditions:

i € c3 (v0)' it is seen that there is an i-edge inddent with v0 which

does not belong to p(v0' u) so that by Lemma 4.1, 1ife have Ag(G) < A.r(G),

as required. Hence, we may assume that c3 (v0) = t.

It is observed that since d(G) = k and the number of' colors used

is k - s, for any vertex v, c2

(v) 1 t, and if' c3(v) = ., then Ic2

(v>l.2: s.

Now, let i l C c2 (v0) and e1 = (v0' vI) be an:r iI-edge. Let us

2 - 2assume, if possible, that (a) c (vI) n c (vO

) = t J, Le., c (vI) c.=. c(vO

),

?- 2 3(b) c (vI) n c ("0) = t, and (c) C (vI) = t hold simultaneously. Then,

f'rom (b), i l <f c2 (vl

) and f'rom (c), Ic2 (vl

) I .2: s. Since there 8,re at

roost s-l edges, other than the il-edgesel , joining V o and VI' it f'ollows

from (a) that f'or some color i 2 e: c2 (vl

), there IlIUl3t be an i?--edge

e2 = (v0' v2)' say, such that v2 f vI· Let us assume f'urther, if'

possible, that c2 (v 2) n c(v0) = t, c2

(v2 ) n c2

(vt ) =t f'or t = 0,1, and

c3(v 2 ) = t. Then, clearly, ~f. c2

(vi) U c2 (v2 ), Ic2

(v l ) u c2(v2)1.2:2S

2 ::>and c (vI) U c (v?-) s. c(v0). Hence, as above, fOJ:" some color

i3

c c2 (v1 ) u c2 (v2 ) there must be an i3

-edge e3

=: (vO,v

3), say, such

that v 3 ~ {v0'v l'v2 }. Continuing the process, it is evident that since

Ic(v0) I :s k-s -1, there must exist an index m, 1:S m :s k-s-l, such that

we C8n f'ind it-edges et = (vO,v t ), t = 1,2, ... ,m satisfying the f'0110;W-

21

In each of the above cases, we shall now obtain a (k-s)-co1oring

2 -c -(vm) n c(vO) f +, or

2 2c (vm) n c (v0) f +, or

c2

(vm) n c2

(vt ) f + for some t, 1 ~ t ~ m-1, or

c3(v ) f +.m

(1)

(2)

(4 )

(iv) C2 (VO)' C2

(V1), •.• , c2

(vm_1 ) are mutually disjoint,

(v) c3 (vo) = c3(v1

) = •.• = c3 (vm_1

) = +,

and further we have one of the following cases:

g of G such that Ag(G) < ~(G).

Case (1). c2 (ym) n c(vo) f t. Let j • c2(vm) n c(vo)' Inter­

changing the colors j and 0, if necessary, we may take j=O. Let the

new coloring of G be denoted by fO' (Obviously, Af (G) = Ar(G), ando

all the conditions (i) to (v) are satisfied with respect to f o.) It

is evident from conditions (ii) and (iv) that there is a (unique)

sequence of vertices

with Vo such that e' f e = e .P1 P1 1

Now, considering the sequence of vertices (4.2), we proceed as

follows: Sin::e i • c2 (yo)' there is an i-edge ep' , say, incidentP1 P1 1

such that the i-edge e and one of the i-edges e and e' doP2 P2 P1 P1 P1

(4.2)

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easily checked that one of these chains must satisfy the requirement.)

Using (II), interchange the colors on ~(vO,uI)' and let f l denote the

and ~'(vO,u') be an (i ,i )-chain with first edge e'. Then, it isPI P~ PI

i-edges e and e' do not belong to --(vO,u2). Using (II), inter-P

2P2 P2 ~c

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Now, consider an (i ,O)-chain p(vo'u),Pa+l

more, a coloring f of G such thata

fa' we have i. E c2

(vo)'Pa+l

(where e = (vO,vp ) is an i-edge).Pa+l a+l Pa+l

22

(Since i € c2

(v ), such a chain can al,vays beP? PI

Vo such that e' f e .Pa+l Pa+l

do not belong to it.

found. In fact, let ~(vO,u) be an (i ,i )-chain with first edge ePI P2 PI

coloring of G so obtained. Evidently, Af~G) ~Af(G), and with respect

to f l , we have i • c2

(vO)' i € c2

(v = ) for j = a+l,a, ••. ,3, andP2 Pj Pj - l

o • c2 (v ) n c(vo)' Since i € c

2 (vO)' there is an i-edge e'm P2 P2 P2'

change the colors on "-2(vo'~) and let f 2 denote the coloring so ob­

tained. Evidently, Af (G) ~ Af (G) ~ A.r(G), and -with respect to f 2,2 I

we have i £ c2

(vO)' i • c2

(v ) for j = a+l,a, ... ,4, andP3 Pj Pj - l

2 ) -(o € c (vm n c vO). Proceeding in this manner, it is obvious that we

Since i E c2 (vo)' there is B.n i-edge e' , say, incident withPa+l Pa+l Pa+1

can obtain, using (II) a-2 timesf

~j (G) ~ Af(G), and with respect toa

=- ? -OEC(V )nc(vo)Pa+l

say, incident with Vo such that e' f e • As above, consider anP~ P2

(i ,i )-chain ~(vO,u2) such that the i-edge e and one of theP2 P3 P3 P3

23

sequence of vertices

i -edges, for j = a+l,a, ..• ,l. Hence, as in case (1), proceedingPj

with the sequence of vertices (4.2), we can obtain g such that

Case (2).

say, with first edge e' • Using (II), interchange the colors onPa+l

~(vO,u) and let g be the coloring so obtained. Since 0 € c2 (v )Pa+l

and ~(vO,u) is minimal, it is seen that e cannot belong to ~(vo,u)Pa+l

so that by Lemma 4.1, we have Ag(G) < ~ (G) S~(G), as required.a

to ~(vO,u), then we take g = 1'0 and by Lemma 4.1, we evidently have

Ag(G) < Ar(G). If both e and e' belong to p(vO,u)(or, equivalently,

if u =vO), then it is observed that with respect to 1'0' we have

o E c2 (v ) n c(vo)' and i € c2 (v ), where e = (vO

' v ) arem Pj Pj - l Pj Pj

by (v), c3(vo) = t, so that there are exactly two i-edges e and e',

say, incident with vO. Let j € c(vo) and consider an (i,j)-chain

~(vO,u), say. Using (II), interchange the colors i and j on ~(vOu),

and then, interchange i and 0 on all edges of G. Let 1'0 denote the(.'

coloring so detained. If one of the edges e and ;i-I does not belong

Ag(G) < ~(G), as required.

2 2Case(~. C (vm) n C (vt ) f • for some t: 1 < t < m-l. Let

i € c2 (vm) n c2(vt ). From conditions (ii) and (iv), there is a (unique)

vt=v ,v, .•. ,v =Vl,v =VO(t=qb+l > qb >...~=O) such thatqb+l qb ql ~

it = i € c2 (v ), i € c2 (v ), ••• , i € c2 (v 1)' i l =i € c2 (v ) .~+l qb qb qb-l ~ q- ql ~

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2c (v),qj-l

seen

In particular, i f i for~

Now, let us first assume that we also have i f i (or,Pr

= a+l,a, •.. ,1. Let j E c(vo), and

say. (Obviously, u f vO.) Using (II),

equivalently, i f i for any r = 1,2, .•. ,t.r

any r = b+l,b, ... ,1.

equivalently, e f e ) for any rPr

consider an (i,j)-chain p(vO'u),

evidently,

where e = (vO'v ) are i -edges, for j = b+l,b, ••. ,l. Hence,qj qj qj

as in case (1), proceeding with the sequence of vertices

interchange the colors on ~(vo,u), and then interchange i and 0 on all

edges of G. Let f O denote the coloring so obtained. Evidently,

Ar (G) SAr(G). Now, if u f v = v , then it is seen that witho m P~+l

2( -) 2respect to f O' we have 0 E C v) n c(vO ' and i E C (v ) wherem Pr Pr - l

e = (vO'v ~'are i -edges, for r = a+l,a, •.. ,l. Hence, as inPr P';~ Pr

case (1), we can obtain g as required. If u = vm' then it is

that with respect to f O' we have 0 E c2

(vt ) n c(vo)' and i .EqJ

Now, in view of the cases (1) and (2), we may assume that ie c(v0 )

and i ~ c2

(vo) so that there is one and only one i-edge e, say, incident

with v. From (ii) and (iv), it is easily seen that e f e or,r

Let j E c(v0)' Interchanging the colors j and 0, if necessary, we

may take j = 0, and denote the new coloring of G by fO

' Now, it is

evident that we can obtain a coloring f r _2 of G, proceeding as in case

(4.3), we can obtain g as required.

Let us now assume that i = i (or, equival,~ntly, that e = e )~ Pr

2for some r = a+l,a, ••• ,1. Since, by assumption, :t~ c (vO

), clearly

i f i • Let i = i (a+l > r > 2), so that i Ii c2 (v ) n c2 (v ).Pl Pr - - Pr m Pr - l

25

with first edge e' and e tl, respectively. Since both of these chains

let f 1 denote the coloring so obtained. In either case, it is seenr-

with v . Consider (i,i )-chains ~'(V ,U') and JltI(v ,uti)Pr - l Pr Pr - l Pr-l Pr - l

Let g be the coloring so obtained.

Now, let e I be an i -edge incident withPr - l Pr - l

to O.

< Ar (G) ~ Af(G), as required. Ifr-2

it is seen that J.lr_l(vO,u ' ) = (v o' ep , tl'{v r_l'u'»r-l

= (vO'v ) are i -edges for j = a+l,a, •.. ,r-l, andPj Pj

E c'2(v ) n c2 (v ).m P 1r-

. 2(). ~( )we have 1 E c VO ' 1 E c v for j = a+l,a, •.. ,r, wherePr - l Pj Pj - l

v 0 such that elf e , and let e I, e tI be any two i -edges incidentPr - l Pr - l Pr

are minimal, it is seen that at least one of them, say JI, (v , U '),

Pr - l

does not contain the i-edge e • If u, f v and u' f YO' then,Pr Pr Pr - l

U I =v l' thenr-

is an (i ,i )-chain which does not contain the edges e andPr - l Pr Pr - l

e • Using (II), interchange the colors on ~r_l(vo'u') and let f r _lPr

(1), such that ~ (G) ~Af (G) = ~(G), and with respect to f r _2,r-2 0

denote the coloring so obtained. And, if u I = V0' then ~ I (v , U I )Pr - l

contains the edge e ' , but does not contain any of the edges e ,Pr - l Pr - l

e and e tI. Using (II), interchange the colors on ~ I (v , U I), andPr Pr-l

change the color of ePr - l

It is easily verified that A (G)g

using (II), first interchange the colors on ~I(Vp ,u'), and thenr-l

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as required.

ed by more than one edge. As an important specil3.1 case of Theorem

we have now a situation similar to case (2). Hem~e, we can obtain g

edges for j = a+l,a, •.• ,l. Hence, as in case (1)" we can obtain g

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i € lev ) for j = a+l,a, ... ,r+l, whereP,i Pj - l

and with respect to f l' life haver-that Af (G).:s ~(G),r-l

2 2i € C (v m) n c (v0),Pr

4. 1, we have the following:

satisfying 6 g (G) < ~(G), as required.

Theorem 4.2. If G is a linear graph which is 10caJ.ly finite,

e = (vo'v ) are i -edges for j = a+l,a, ... ,r. It is evident thatPj Pj Pj

Case (4). c3 (v ) f t. Let i E c3(v ). From the above cases_______m___ m

(1), (2), and (3), we may assume that ie'c(vO), it c2 (vo) and

2 -i. c (v t ) for any t = 1,2, ••• ,m-l. Hence, there is one and only one

i-edge incident with v 0' and i f it for any t = 1,,2, .•. ,m. Let

j c c(vo). Consider an (i,j)-chain ",(vO,u). Using (II), interchange

the colors on p(v0' u), and then, interchange i anel 0 on all edges of

G. Let f o denote the coloring of G so obtained. It is clear that

Af (G) .:s 6r(G), and furthermore, with respect to :f'0 we haveo

o e c2

(v ) n c2

(v0), i € c2

(v ), where e = (vO,v ) are im Pj Pj - l Pj Pj Pj

Thus, we have shown that starting with any (k-s)-coloring f of g

such that Ar(G) > 0, we can obtain a (k-s)-coloring g of G satisfying

Ag(G)<ty(G). The proof of the theorem is n01v completec1 by finite induction.

A graph is said to be linear if no two of its vertices are join-

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then k(G) = d(G) or k(G) = d(G)-l. For any integer k ~ 2, there exist

linear graphs G such that d(G) = k and k(G) = d(G)-l.

It may be remarked that the proof of the inequality k(G) ~ d(G)-s

in Theorem 4.1 is algorithmic so that given any finite s-graph G we

can find a partition of its set of edges into d(G)-s sets each of which

is a cover of G.

Remark: We state here without proof that the Theorem::3.1 and 4.1

remain valid if we replace the condition of locally finiteness by the

more general conditions: (a) d(G) = k < • and (b) each connected

component of G has an enumerable number of vertices. The above

theorems may hold for arbitrary graphs satisfying only condition (a).

However, no proof or counterexample to it is known to us.

5. Duality between Cover Index and Chromatic Index of a Graph

Let G be an undirected graph. Any subset M of E(G) is called

a matching of G if each vertex v € V(G) is incident with at most one

edge in M. The chromatic index of G, denoted by q(G), is defined to

be the minimum number q for which there exists a partition of E(G)

into q sets each of which is a matching of G. Equivalent~, the

chromatic index q(G) of G may be defined, as usual, to be the minimum

number q for which there exists a q-coloring f of G such that for

each vertex v E V(G), there is at most one i-edge incident with v for

i = O,l, ••• ,q-l. It may be observed that the concept of the cover

index of a graph, as introduced in this paper, is apparently dual to

the concept of the chromatic index of a graph, and it is interesting

to compare the knmm results concerning these two concepts. We may,

28

known, however, if the converse implication also holds.

therefore, add the following remarks:

ous1y. Also, it would be desirable to have crite~ria under which the

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(The inequality q(G) < d(G) + s. -Very little is known in this respect.

(1) We proved (Theorem 3.1) that if G is l'l. (locally finite)

bipArtite graph, then keG) = d(G), and as is well-known, vie also have

q(G) = d(G) where d(G) is the least upper bound of the degrees of the

vertices of G. In fact, the equality q(G) = d(G) is equivl'l,lent to

the theorem of Konig, Corollary 3.3, (see, for instance, C. Berge [1],

p. 99), and hence is implied by its counterpart keG) = d(G). It is not

iTl

is tempting to ask if anyone of the~ CM be derived from the other.

expect, such a relationship does not hold in genElral. For instance,

was initially proved by V. G. Vizing [12], and l~Lter rediscovered by

the author independently. See, e.g., [4,5J.)

(3) It can be easily seen that if G is a regular graph, then,., -

keG) = d(G) if and only if q(G) = d(G). Unfortunately, as one might

if G is the graph of Figure 1, then keG) = d(G) 'but q(G) = d(G) +1,

and if G is the graph of Figure 2, then q(G) :: d(G) but keG) = d(G)-l

In general, for any s 2 1 and any fixed r, 1:S r :s s, one can construct

examples of s-graphs G such that keG) = d(G) where q(G) = d(G) + r

and also, examples of s-graphs G such that q(G) := d(G) whereas

keG) = d(G) - r. Hence, one can only ask for SOmE! criterion under

which both the equalities keG) = d(G) and q(G) = d(G) hold simu1tane-

(2) We know that "if G is any locally finite s-graph then

d(G) :s q(G) :s d(G) + s; the bounds are best possible in the following_'.. (, .1,

- ]~+l :.:cases: s 2 1 and d(G) = 2ms-r where r 2 0, m > ~~J. Clearly, the

I'l,bove theorem and Theorem 4.1 are counterparts of each other and it

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equality k(G) =d(G) or q(G) = d(G) holds.

Figure 2.

q(G) = d(G), k(G) = d(G) - 1Figure 1-

k(G) = d(G), q(G) = d(G) + 1

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ACKNOWLEDGEMENTS

}bst of the present research work was carried at the Tata

Institute of Fundamental Research, Bombay and is embodied in the thesis

of the I'l.uthor [5]. It is a grep,t pleasure to express my sincere

gratitude to Professor S. S. Shrikhande for supervising the research

work and providing steady encouragement in the :preparation of the

mI'l.nuscript.

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REFERENCES

[1] C. Berge, Theory of Graphs and its Applications, Methuen and

Co., London (1962).

[2] R. P. Gupta, A Decomposition Theorem for Bipartite Graphs

(Abstract), Theory of Gre,phs, International Symposium,

Rome, July 1966, pp. 135-136.

(3) R. P. Gupta, A Theorem on the Cover Index of an s-gr!'l,ph

(Abstract 637-14), Notices of the Amer. Math. Soc., Vol.

13 (1966), p. 714.

[4] R. P. Gupta, The Chromatic Index and the Degree of a Graph

(Abstract 66T-429), Notices of the Amer. Math. Soc.,

Vol. 13 (1966), p. 719.

[5] R. P. Gupta, Studies in the Theory of Graphs (thesis), Tata

Institute of Fundamental Research, Bombay (1967),. Chapter 2.

[6] P. Hall, On Representations of Subsets, Jour. London Math.

Soc., Vol. 10 (1934), pp. 26-30.

[7] D. ~nig and S. Valko, Uber Mehrdentige Abbildungen von Mengen,

Math. Annalen, Vol. 95 (1926), pp. 135-138.

[8] D. KOnig, Theorie der endlichen und unendlichen Graphen,

Akad. Verl., Leipzig (1936), pp. 170-175.

[9] o. Ore, Theory of Graphs, A-M-S. Colloquium Publications 38

(1962), Theorem 13.2.3, pp. 208-210.

[10] o. Ore, loco cit., Problem 4, p. 210.

[11] J. Petersen, Die Theorie der Regularen Graphs, Acta Math.,

Vol. 15 (1891), pp. 193-220.

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[12] V. G. Vizing, On an estimate of the chromatic class of a p-graph

(Russian), Discrete Analiz, No.3 (196!J.), pp. 25-30.

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