Abh. Math. Sem. Univ. Hamburg 56, 127--137 (1986)

Longest paths joining given vertices in a graph

By H. A. Ju~G

dedicated to K. Wagner on his 75 th bir thday

1. Introduction

The graphs in this paper have neither loops nor multiple edges. For a graph G let V(G) denote its vertex set, ]G I the cardinality of V(G) and d(v) the valence of the vertex v in G.

The following result on longest paths joining specified vertices in a graph is due to S. C. LOCKE [7].

Theorem 1. For distinct vertices a, b and c in the 3.connected graph (7 there exists an ab.path P through c such that IPI = IG 1 or FPI ~ 2d(v) -- 1 /or some v ~ V(G).

An Ore-type theorem on longest ab-paths was proved by H. E~OMOTO [3].

Theorem 2. Let a and b be distinct vertices and P a longest ab-path in the 3.connected graph G. I / ] P ] ~ JG l there exist non-adjacent vertices u and v in G such that IPl ~ d(u) -~ d(w) - - 1.

The purpose of this paper is to show that under certain restrictions better bounds for the length of longest ab-paths can be obtained.

Paths in G are considered as subgraphs of G. For a path P in G let (7 -- P denote the maximum subgraph of G spannend by V(G) - - V(P), and for a component H of G - - P let pH denote the maximum subpath of P whose terminal vertices are joined to H. A subgraph H of G is called a stronghold in G if it is a complete graph and any two vertices of H have the same neighbors in G outside H.

Theorem 3. Let a and b be distinct vertices and P a longest ab-path in the It-connected graph G where 2 ~ k ~ 5. Each component H o / G - - P with [HI ~_ k - - 2 contains a vertex v such that IPHI ~ (]c - - 1) (d(v) - - ]c + 2) ~- 1; moreover ]PH l ~ (]r - - 1) (d(v) - - k ~- 3) ~- 1 unless H is a stronghold in G with IHt <_ 2k - 4 or INp(H)I = k.

The case k ---- 2 was first considered by P. ErdSs and T. Gallai (see Theorem (1.16) in [4]). For/c ---- 3 a corollary is proved (Corollary 2.2), which yields tha t the estimate IPI ~ d ( u ) ~ d ( v ) - 1 in Theorem 2 can be replaced with IP[ ~ max (2d(u), 2d(v)) -- 1.

By Theorem 3 a longest ab-path P in a 4-connected graph G is either domi- nating (i.e. G -- P is independent) or has at least 3d(v) -- 5 vertices for some v c V(G) - - V(P); the estimate [PA I ~3d(v) -- 2 when H is no stronghold

128 H.A. Jung

in G will be needed in a projected paper about longest ab-paths in regular graphs.

To see tha t the factor 4 cannot be increased by increasing the connectivity consider the disjoint union H ~ H1 u H2 u ... u / / , where n ~ k ~ I and each Hi is a K3 or a star on at least 3 vertices; add vertices Vl, v2, ..., vk and join each vi to each v E V(H). The construction gives a graph G of connectivity k and minimum valence k ~- 1 (if at least one Hi is a star). A longest vlv2-path in G has 4k - - 3 vertices.

Notice tha t Theorem 3 also holds for vertices a and b in an infinite graph provided a longest ab-path exists.

Simultaneously we prove the following result.

Theorem 4. Let a and b be distinct vertices and P a longest ab-path in the k-connected graph G where k ~ 3 or k ~ 4. Each component H o / G -- P with IHI ~_ k -- 1 contains a vertex v such that [G] ~ k(d(v) -- k ~- 2), moreover [G] ~ k(d(v) -- k -~ 3) i / H is no stronghold in G.

Related results on longest circuits were proved in [8], [1] and [6]. For other results on longest ab-paths see S. C. LOOK~ [7], J. A. BONDY and B. JACK- so~ [2].

2. The induction step

For vertices a' , b' on a pa th P let P[a', b'] denote the subpath of P with terminal vertices a ' and b'. For subgraphs K and L of G let NK(L) denote the set of all vertices in K which are adjacent to some vertex in L. For a subgraph K of G and v E V(G) abbreviate dK(v):----INK(V)[. Let u(G) denote the connec- t iv i ty of G.

In the proof of the following result we also set up the notation for the later proofs.

Theorem 2.1. Let a and b be distinct vertices and P a longest ab-path in the 2-connected graph G. Each component H o / G -- P contains a vertex v such that

(i) IPHI ~ d(v) ~- 1 (ii) ]P~[ ~ d(v) ~- 2 i] H has a cut vertex (iii) IPH[ ~ 2d(v) - - 1 or [HI ~ d(v) -- 1 and u(G) = 2.

Proof. By induction on [P[. Let H be some component of G - P and label Np(H)----{x 1, x 2 . . . . . xs}

according to the order on P from a to b. Since P is a longest ab-path, xi+l is not the successor of xi. Set I : {i: ]NH(Xi, xi§ ~ 2}.

I f I = g then NH(P) = {v} for some v E V(H) and hence V(H) ~-- {v} since G is 2-connected. I f V(H) ---- {v} then ]P~[ _~ 2s - - 1 ---- 2d(v) - - 1.

For i s I determine a longest pa th Qi = Qi[ui, wi] in H such tha t u~ E NH(Xi) and wi E •g(Xi+l). Clearly such a pa th exists and ]P[xi, xi+1]] ~ 2 ~- ]Qi[ since P is a longest ab-path. Notice tha t Q~ is a longest uiwi-path in H. Pick j E I such tha t ]Qil ~_ [Qi] for all i E I .

Set X ---- {xi: dH(xi) ~ 2} and Y --~ {v E V(H): N1,-x(v) ~= 0}. Clearly X u Y is a cut set of G unless V(H) -: Y. Also i E I u {s} if xi E X. I f y E Y there is some

Longest paths joining given vertices in a graph 129

xi C Np-x(y) such that s ---- i or x~+l r N~,-x(y) hence i E I u {s}. Therefore

Ix u y[ _< ]I] + 1. (1)

Case 1. H has 11o cut vertex. If V(H) = {ui, wi} and @(ui) < s then [pHi _~ 2s ~ 2d(ui) . If V(H) = {u i, wi} and dp(ui) = dp(wi) ---- s then I = {1, 2, ..., s -- 1} and

]pB]_~3s_2=3d(ui)_5. In the latter subcase [ P ~ [ ~ 2 d ( u i ) - - I or d(ui) = d(wi) = 3 = s -~ 1.

Thus let ]HI _~ 3. By induction hypothesis there is some vl C V(H) -- {u i, wj} such that IQj[ -~ dH(v~) § 1.

If ]I] ~ 2 then [pHI ~ 2s -- 1 + 2 ( ] Q i I - 1 ) ~ 2 d p ( v j ) - l + 2 d H ( v j ) = 2d(v) -- 1.

Assume ] I ] = 1 . By (1) we have [Xu Y I ~ 2 , in particular [ Y ] ~ [ H [ , hence X u Y is a 2-element cut set of G. Also hrp(vj) C {xi, xi+l} since I = {?'}. Therefore

[P ' ] ~ 2 + IQ~] ~ 3 + dH(vj) ~ d(v~) + 1 and

IH[ = 1 + d.(v~) + d . (v j ) ~ d(vj) -- 1 + d . (v i ) .

Case 2: H has cut vertices. Determine a longest path Q0 = Qo[cl, c2] in H whose terminal vertices are

cut vertices of H. If such a longest path should not exist (which will be dis- proved) determine a path Qo = Qo[cl, c2] with cut vertices cl, c2 of H and ]Qo] _~ 2 [P[.

Further determine distinct components K1 of H - Cl and K2 of H - c2 such tha t K1 and K2 contain no vertex of Qo. Take the choice of K1 and K, so that ]K1] + [K2[ is maximum. Clearly Np(Ka) ~= 0 for h = 1 and 2 since G is 2-connected. Label ]gv(K1 u K,) = {zi, z2 . . . . . Zq} according to the order on P from a to b.

First assume q _~ 2. There is some j < q such that zj C Np(K1) and zi+ 1 C N~,(K2) or else z i E .Np(K~) and zi+ 1 ~ IYp(K1), say w~ ~ NK,(zi) and w2 ~ NK~(Zi+~). l~or paths Q1--Q~[w~, c~] and Q2 = Q~[w~, c2] in H one obtains ]P[zi, zi+~] [ _~2-}-[Q0 ~Q~ ~ Q2[ since P is a longest ab-path. In particular [PI ~ ]Qo] hence Qo in fact is a longest path in H whose terminal vertices are cut vertices of H. One deduces that K~ ~ K~ contains no cut vertex of H. Also, for h -- 1 and 2 there exist longest waca-paths Qa in H and ]P~[ ~ 3 -}- ]Qa[.

If [Kal ---- 1 thent rivially [Qa[ -- d~(v~) + 1 ---- 2 where V(Ka) ---- {va}. If ]Ka] _~ 2 the maximal subgraph K~ of H spannend by V(K~) ~ {ca} is 2- connected, and, by induction hypothesis, there is some va ~ V ( K a ) - {wa} such that ]Q~] ~ dK.(v~) -k 1 = dg(va) q- 1 (h = 1 and 2). Thus

[P~[ ~ 2q -- 1 + ]Q0 v Q~ ~ Q2] - 1 ~ d~(Vl)

-J- dp(v2) - - 1 -~- dH(Vi)-~- dH(v~)

hence ]P ' ] ~ d(v~) ~- d(v~) ~ I with strict inequality unless d~(v~) = d~(v~) = q (in which case d(v2) ~ 3 and IP~l ~ d(v~) + 2).

130 H.A. Jung

Final ly assume q = 1. P ick wa ~ NK~(z~) and notice t h a t {ca, z~} is a cut set of G for h = 1 and 2. Since G is 2-connected there is some z ~ Np(H) - - {Zl}. P ick v ~ N~(z) and de te rmine a p a t h R ~-- R[v, w] in H such t h a t R ~ Q0 ~- w. Fo r pa ths Q~ = Q~[w~, c~] and Q~ = Q~[w~, c2] in H one obta ins ]P[z, z~][ ~ _ 2 § in par t icu lar [P~[ ~ 2 § ( h : 1 , 2 ) and

2 I P ' l _~ 6 § lQo[c~, w][ + [Q0[c~, w][ = 7 + [Q0[.

Hence, in fact Qo is a longest p a t h in H whose t e rmina l vert ices are cut vert ices of G, and we m a y assume t h a t Qa is a longest waca-path in H. As in the case where q ~ 2 one can f ind vert ices va ~ V(Ka) such t h a t IQh[ _~ dn(va) § 1 (h = 1, 2). Hence

IP ' I ~ 2 § IQ~ ~ Qo[c~, w] ~ R[ ~ 3 § d.(va) ~_ 2 § d(va), (h = 1, 2).

Also If-/[ ~ d~(Vl) § dH(v2) + t ~ d(vl) + d(v~) -- 1.

The reasoning in the proof of the following corollary is similar to t h a t used in the proof of the analogous result for longest circuits [5].

Corollary 2.2. Let a, b be distinc~ vertices and P a longest ab-path in the k- connected graph G, where k ~ 3 and [P4 < [G[. There exist independent vertices vl, v2, ..., vk in V(G) -- {a, b} such that [P[ ~ 2d(vi) - - 1 / o r 1 H i ~ k -- 1 and IPI ~ d(vk_~) § d(vk) -- 1.

Proof. F i rs t assume Np(H) ~ {a, b} for some componen t H of G - - P , say b r Np(H) . Set N p ( H ) = {xl, x2 . . . . xs} and let us be the successor of x~ ( l ~ i ~ s ) . If NG-e(ui)~:~ pick a componen t H~ of G - - P such t h a t ut ~ Np(Hi). B y Theorem 2.1 there is Zo E V(H) such t h a t [PI ~ 2d(zo) - - 1 and zi ~ V(H~) such t h a t IP[ ~ 2d(z~) - - 1.

Set S = {Zo} u {zi: dG_p(U~):~ 0} u (us: de_p(Ui)-~ 0}. Since P is a longest ab-path , for i ~: ~" vert ices us, u i are not adjacent , and H i ~: H i (if defined) ; also H ~ H~ whenever da-p(ui) :# O. Hence S is an independent set and

The following s t anda rd a rgumen t yields IP[ _~ dp(u~) § de(ui) - - 1 for i < j. I f x 6 Np(ui) where x lies on P[ui, u i] t hen the predecessor of x is not ad jacen t to u i since P is a longest ab-path.

Similar ly if x E Np(ui) where x lies on P[a, us] or P[uj, b ] - b t hen the successor of x is not ad jacen t to u i. Hence de(ui) ~ dp(ui) - - 2 and [Pr : 1 § @(uj) § dp(ui) ~ dp(uj) § de(us) -- 1

Relabell ing {us: dG_p(u~) = 0 / : : {vl . . . . , %} so t h a t d(Vl) ~ d(v~) ~_ ... ~_ d(v~) yields [P[ ~ d(vi) § d(vi+i) - - 1 ~ 2d(v~) - - 1 for 1 ~ i ~ t - - 1.

FinaUy assume N~(tt) ~ {a, b} for all components H of G - - P . Le t x be the last ve r t ex yon P - - b such t h a t Ne_r(x ) ~= ~. Pick a componen t H of G - - P with x ~ Np(H). Wi th the above no ta t ion for this componen t H one s imilar ly obta ins an independent set

S = {z0} ~ {z~: d~_~,(u~) :# 0} v {u~: d~_e(u~) ---- Ol

Longest paths joining given vertices in a graph 131

and IS] = 1 + s - - 1 ~ lc. B y assumption, ~VG_P(Ul) = ~ and Na_e(ui) ---- 0 where u i is the successor of x = x i. Now the claim follows by the argument of the last paragraph (since t _~2).

Corollary 9..3. Let a and b be distinct vertices and P a longest ab-path in the 2-camnected graph G o[ m in imum valence d. I f IG] ~ 3d - - 2 and IP] < [G I then

(i) IP] ~ 2 d - - l o r (ii) {a, b} is a cut set o[ G and G has a hamiltonian circuit.

Proof. I f G - - P has two components H1 and H2 one m a y assume, by (iii) in Theorem 2.1, tha t ]HaT ~ d(vh) - - 1 for suitably chosen vertices va E V(Ha) (h = 1 and 2). Since [PI ~_ d(vo) + 1 for some Vo C V(G) one obtains 1(71 ~ d(vo) + d(vl) + d(v2) - - 1.

Let H be the only component of G - - P and Np(H) ~-- {xl, x~ . . . . . xs}. I f xl ~ a or b ~= x2, say b ~= x2, consider the successors ua of xa(h ~ 1, 2) : one obtains IPBI ~ d(ul) ~ d(ua) - - 1 (see proof of Corollary 2.2) hence the claim.

Thus let Np(H) -~ {a, b}. I f H has a cut ver tex zl then IHI ~ dH(vs) -~ d~(v2) + 1 ~ d(vl) + d(v2) - - 3 for vertices vl, v~ from distinct components of H - - zl, further ]Pa I ~ d(vo) + 2 for some Vo C V(H).

Let H have no cut vertex, and let Q = Q[a', b'] be a longest pa th in H such tha t a' C NR(a) and b' C NH(b). I f [QI : IH] we are done. Thus let K be a component of H - -Q . By Theorem 2.1 there is a ver tex v in K such tha t IQgI ~ 2dH(v) - - 1 or else IQKI ~ dH(V) ~- 1 and Ig] ~ dH(v) - - 1.

Label NQ(K) ~-- {Yl . . . . . Yt} according to the order on Q from a' to b'. If v E s then Yl =~ a ' since otherwise one would obtain a pa th R -~ R[v, b'] ~ Q in H contrary to the choice of Q. Similarly v r NH(b) or Yt ~ b'. Therefore IQI ~_ ]QKI -~ alp(V).

I f IQ K] ~_ 2dH(v) - I then [ Q l - - ~ 2 d H ( v ) ~ - d p ( v ) - l ~ 2 d ( v ) - - 3 hence IP] _~ 2 § IQI ~- 2d(v) - - 1. Else IQI ~ d~(v) § 1 + dp(v) = d(v) § 1, [Pl ~-- 2 -~ ]QI ~ d(v) -b 3 and IKI ~ d~(v) - - 1 ~ d(v) - - 3.

3. Proof of Theorems 3 and 4

In this section a 2-connected graph G, distinct vertices a and b, further a longest ab-path P in G and a component H of G - - P are fixed.

Lemma 3.1. Let G be it.connected where 2 ~ lc ~ 5. Further let [Ht ~ lc - - 2 and H have no cut vertex. Then IPH! ~ (lc - - 1)(d(V) - - lc -t" 2) ~- 1 for some v C V(H); moreover IP'] ~_ (]c - - 1) (d(v) - - lc + 3) -~ 1 unless H is a stronghold in G.

Simultaneously we prove the following lemma.

Lemma 3.2. Let G be it-connected and [HJ ~ Ic - - 1, where lc ~ 3 or k ~ 4. I] H containz no cut vertex theu IGI ~ k(d(v) - - lc + 2 ) / o r 6ome v C V(H); moreover ]q I ~_ lc(d(v) - - lc + 3) unless H is a stronghold in G.

132 I-I. A. Jung

Proof. Le t N~(/ / ) = {xl, Xz, . . . , x,}, I and Q~ = Qi[ui, wi] for i E I have the same meaning as in the proof of Theorem 2.1.

Also JQ~J > ]Qj] for all i E I . I f I = g then NH(P)={v} hence V( t t )={v} ; if V(H)----{v} t hen IPH]

>_ 2d(v) - - 1. Thus assume I ~ ~3 and ]H I > 2. Given dist inct vert ices vl, v2 E V(H) the es t imates IP[x~, X~+l]] > 2 3- [Q~[ for

i E I yield

1PHI ~ 2dp(vl) 3- 2d~(v2) - - 2t - - 1 3- l I [ . (]Qi] - 1) (2)

where t : INp(vl) n Np(v2)l.

We nex t choose vert ices vl and v2. I f (i) H is complete let vl, v2 be a rb i t ra r i ly chosen in H. I f (ii) H is not complete and 1HI -~ ]Qi] let vl and v2 be non- ad jacen t vert ices in H.

I f (iii) H - - Qj has dis t inct componen t s K 1 and K2 pick vh E V(Ka) such t h a t IQ~ h] ~ dH(Vh) 3- 1 (by Theorem 2.1 for h : 1 and 2).

I f (iv) H - - Q i has only one componen t K1 choose vl E V(Ka) with IQ~'[ > dg(Vl) 3- 1 b y Theorem 2.1, and let v2 be the successor of the f irst ve r t ex on Qi in N~,(K1).

I n all cases ( i )--( iv) one has [Qi] -~ dH(Vh) 3- 1 for h ~-- 1 and 2. Case 1: ]I] _~ 4 and 1Qi] ~ 3.

I n view of [Qi] -~ dH(vh) 3- 1 and dp(vh) 3- dH(vh) ~-- d(vh) for h ---- 1 and 2 one obta ins b y (2)

] P ' ] = 2d(v~) 3- 2d(v~) 3- (]I] - - 4) (IQjl - - 3) 3- 2(111 - t) - - 9 + (~ (3)

where 0 ~ 0. Not ice t h a t ]PH 1 ~ 4d(vh) - - 7 implies [pH] ~ 3d(vh) - - 2 and tP H] ~ 2d(Vh)

3- 3 unless d(vh) ~ 4; if d(vn) ~ 4 t hen IPH[ ~_ ]I I (]Qj.] 3- 1) 3- 1 ~_ 17 yields I P ' ] > 4d(va) 3- 1. Also t 3- IY[<_ IX t 3- IY] < ]II 3- 1 b y (1).

Case 1.1. xt q Np(Vl) n Np(v2) for some l, 1 < 1 < s. One m a y assume 1 = s or xl+x E Np(vl)n Np(v2). Then 1 E I u {s} hence

III _> t. I f xt r Nl,(vl) u Np(v2) or ]Qi] ~> dH(Vn) -}- 2 (h ---- 1 or 2) then ~ > 2 in (3) hence the claim.

Now suppose [Qi] = dH(vl) 3- 1 ----- d/t(v2) 3- 1 and xt E Np(vx) u Np(v~), say x~ c N~(va).

I f x~ E Np(H) -- Np(vl) for some m, we can pick m so t h a t m ---- s or Xm+x r Np(tt) -- Np(vl). I n t h a t subcase m =~ l and m E I u {s} hence ]I] ~ t q- 1 and the claim b y (3). I n the remain ing subcase one hase x~ E N~(Vl) for all i whence d(va) > d(v~) and [ P ' ] ~ 2d(va) 3- 2d(v2) - - 9 ~ 4d(v~) ~ 7.

Case 1.2. xiENp(v~)nN~(v2) for all i, l ~ i ~ s . Then l l l - - t - l . I f ]Qi] ~ dH(va) 3- 2 for h ---- 1 and 2 t hen ~ ~ 4 in (3) hence the claim. This covers case (ii). I n cases (iii) and (iv) l abe l 2VQ~(K1) ----- {y~, ..., yq} according to the order on Qi f rom u] to w i. Now yq :4= w i since otherwise Qi and a p a t h f rom w~ to v~ with inner ver t ices in Ka would define a p a t h R i ---- Ri[ui, v~] in H . B u t IRit ~> IQil, ul E N~(xi) and va E NH(Xi+a ) imp ly a contradic t ion to the choice

Longest paths joining given vertices in a graph 133

of Qi" Therefore [Qil > IQ~'I > dH(Vl) + 1. Similarly 1Qjl >_ dH(v~) + 2 in case (iii).

I n case (iv), since Qj is a longest u~,w~-path in H, ve r t ex vz is no t ad jacen t to the successor, of yq hence [Q~I ~> dH(v2) § 2.

I t remains to consider case (i). I f x~ r Np(v) for some i and some v E V(H) t hen d(v) < d(vh) for h = 1 and 2 hence IPH] > 2d(vl) + 2d(v2) - - 11 > 4d(v)

- - 7. I f v C Np(x,) for all i and all v C V(H) t hen H is a s t ronghold and IPH I > 2d(Vl) + 2d(v2) - - 11 b y (3).

Case 2: [Qj[ = 2. Then [HI = 2 since IQil >- dH(vl) + 1 and H has no cut ver tex . Assume

dj,(v~) > dp(v2). Inequa l i t y (2) yields IPHI > 2d(v~) + d(v2) -4- dp(v2) - - t

4- (]II - - t) - - 4. I f dp(v2) > t, there exist x~ E Np(vl) - - Np(v2) and xm ~ Np(v2) - - Nv(Vl).

One m a y assume l = s or x m r 2Yp(vl) - - Nv(v2) hence 1 ~ I u {s}. Similar ly one m a y assume m ~ I u {s} hence [I I > t + 1 and ]P HI _> 2d(vl) -4- d(v2) - - 2.

I f dp(Vl) > dp(v2) = t there is k ~ I u {s} wi th xk E N~(Vl) - - hr~(v~). I n t h a t subcase ]I I > t and [PHI _>_ 2d(vl) -4- d(v2) - - 4 > 3d(v2) - - 2.

I n the remaining subcase one has d~(Vl) = dp(v2) = t whence H is a s t rong- hold in G and ]PH I > 3de(v1) - - 5.

Case 3: [I 1 = 3 and ]Qjl > 3 . Then t -4- ]Y] <~ 1Zl -4- ]Y] <~ 4 by (1).

Case 3.1: V ( H ) = Y. I f [HI = 4 t hen t = 0 and IPRI > 2d(vl) -4- 2d(v2) - - 7 b y (2). I f IHI = 3 t hen t < 1 and [P~] > 2d(v~) -4- 2d(v~) - - 5 b y (2).

Case 3.2. V(H) :# Y. Then k < 4, since X u Y is a cut set of G. F r o m (2) deduce

IPHI > 2d(vl) -4- d(v~) -4- dp(v2) - - 2t - - 1 and (4)

]P~l > 2d(v2) -4- d(v~) -4- d~(v~) - - 2t - - 1.

I f dj,(vl) _>_ dj,(v~) > t t hen t < 2 as in ease 2 hence IPHI > 2d(vl) -4- d(v2) - - 2 by (4). Also IGI > 2d(Vl) -4- 2d(v2) - - 4 since IH[ > dH(v~) -4- 1. Nex t assume dj,(vl) > dl,(v2) = t. Then there is some 1 C I u {s} such t h a t xt ~ N~(vl) - -Np(v2) hence t < 3 fur ther d(v~) > d(v2) or dH(V2) > d~(vl). I f d(vl) > d(v2) t hen ]P'I > 2d(Vl) -4- d(v2) - - 4 >_ 3d(v2) - - 2 and [G[ > 2d(v~) -4- 2d(v~) - - 6 > 4d(v~) - - 4 b y (4). I f dH(V2) > dH(Vl) t hen ] P ' ] > 2d(v2) -4- d(Vl) -4- dp(Vl) - -2 t b y (2) since IQi[ > dH(Vl) -4- 2.

The subcases in which d~(Vl) < dp(v2) are symmet r ic . I t remains to consider the subcase in which d~(v~) = dp(v~) = t. I f Np(H) =4= N~(v~, ve) there is some x~ ~ N~(Vl, v2) such t h a t Xt+l ~ Np(v~, vz) or 1 = s. Then again t < 3, fu r ther ]P~] > (2alp(v1) -4- 2d~(v~) - - 2t -4- 2) - - 1 -4- 3(IQ~[ - 1) > 2d(v~) -4- d(v~) - - 2 and ]G[ > 2d(v~) -4- 2d(v~) - - 4. I f Np(H) = Np(v~) =- N~(v~) and (i), bu t N~(v) :# IVy(H) for some v ~ V(H) t hen 1P H] > 2d(vl) -4- d(v~) - - 5 > 3d(v) - - 2 and IOl > 2d(v~) -4- 2d(v2) - - 8 > 4d(v) - - 4 by (4). The a r g u m e n t in Case 1.2

134 H . A . Jung

yields [Qj] _>_ dH(Va)-% 2 (h = 1 and 2) in cases (ii), (iii) and (iv). I n those cases [P~] > 2 d ( v t ) - % d ( v 2 ) - - t + 2 and IG] > _ - - 2 d ( v l ) + 2 d ( v 2 ) - - 2 t - % 4 b y (2).

I f H is a s t ronghold in G, then [PHI __>~2d(vi)-~-d(v2)--5 and IGI ~ 2d(vi) -% 2d(v2) - - 8.

Case 4: llI = 2 and [Qi] > 3.

Then t -% [Y[ g IX] + I Y[ _= 3 by (1). I f V(H) = Y then t = 0 and [H] = 3 hence IPB[ =~ 2d(vx) + 2d(v2) - - 5 by (2).

Thus assume V(H) =4= Y whence X u Y is a cut set of G. Le t I ~-- {]', l} and wi thou t loss of genera l i ty j < 1. I f 1 < ] t hen N/~(xl, x2 . . . . , xi) = {ui} and if l + l < s then N~(Xl+l, Xt+2 . . . . , x s ) = {wt}. I f ] - % 1 < l then NH(xi+I, xj+~ . . . . . xt) = {wi} = {u~}.

Case 4.1. Le t vl, v2 be chosen according to (iii) or (iv). Then {Vl, v2} n {ui, wi} = 1~ b y construct ion. Also vh =~ wt for h - - 1 or 2 hence Np(v~) ~ {xi, xi+l, xl+l} wi th s t r ic t inclusion unless s = 3.

Therefore I//I _> 2 + dx(va) ~ d(v~) - - 1 .

Since IP~l _~ 2s - - 1 § 2([Qj[ - 1) _~ 2de(vh) - - 1 + 2dR(va) = 2d(va) - - 1 it remains to consider the subcase in which s = de(vh) t h a t is s : 3 and Ne(H) ---- Np(vh). Also [PB I ~_ 2s - - 1 + 2d~(vl) _~ 2d(vl) -% 1 unless dp(vl) = s. Assuming dp(vl) = s one has in par t icular xj, xi+ 1 ~ Ne(vl). B y the choice of Q1 one obta ins IQ][ > [QI~I[ > dtt(vl ) -% 1 (see Case 1.2).

Now IPHI > 2de(v1) -% 2(IQil - 1) - 1 _> 2d(Vl) -% 1.

Case 4.2. Le t vl, v2 be chosen according to (i) or (ii). Then IHI --= [Qil hence IPHt > 2s - - 1 -% 2([HI - - 1). P ick Vo ~ V(H) -- Y. Then Ne(v0) ~ {x i, x~-+t, xt+l} wi th str ict inequal i ty unless s = 3. I f dp(vo) <. s t hen IP HI > 2d(v0) -% 1, moreover de(vo) < 3 hence ]HI > 1 + di~(Vo) >__ d(vo) - - 1.

Assuming dp(Vo) = s one obta ins s ---- 3. I f de(v) < 3 for some v e. V(H) t hen ]phi > 2d(v) -% 1 and IH] _> d(v) - - 1. Le t de(v) = 3 for all v E V(H). I f d,(v) + 1 < [H I for some v E V(H) t hen [H I > d(v) - - 1 and IPHI > 2dp(v) - - 1 + 2(dn(v) q- 1) = 2d(v) + 1.

I n the remaining subcase H is a s t ronghold in G and [PH I >_ 2d(v) - - 1 for v c V(H).

Case 5. II I = 1 and ]Qi[ ~ 3.

Then IX u YI ~ 2 and X u Y is a cut set of G. Fu r the r [PHI ~ 2S - - 1

% l Q i ] - - l ~ s + d e ( v a ) - % d H ( V h ) - - I for h = l and 2. Therefore IPHI d(vh) -% 2 for h --~ 1 or 2 unless s ---- 2, dp(vl) = de(v2) = s and [Qjl - - dH(vl)

+ 1 = dH(vd + 1.

Now d~,(vl) = dr(v2) = s and [Qi[ = dH(Vl) + 1 imply t h a t H is complete as shown in Case 1.2. I f Np(v) =4= NI,(H) for some v 6_ V(H) and dp(vi) -~ dp(v2) = t hen IP n] > d(v~) -% 1 ~> d(v) + 2. I n the remaining subcase H is a s t ronghold and IP ' I > d(vx) + 1.

The proof of L e m m a 3.1 and L e m m a 3.2 is complete.

Longest paths joining given vertices in a graph 135

Lemma 3.3. Let G be 3-connected and let H have a cut vertex. There exists a vertex v in H such that

(i) ]P ' [ ~ 2d(v) § 1, (ii) IP ' l ~ 3d(v) - - 2 or IHI ~_ d(v) - - 1,

(iii) IP ' I ~ 3d(v) -- 2 and IPHI ~ 4d(v) -- 7 i / G is 4-connected, (iv) IP~I ~ 4d(v) -- 5 if G is 4-connected and H is not a star.

Proof. Determine Qo ~ Qo[Cl, c2], K1 and K2 as in the proof of Theorem 2.1. I t turned out tha t Q0 is a longest path in H whose terminal vertices are cut vertices of H. Thus K1 u K2 contains no cut vertex of H. Fur ther Np(Kh) ~ 0 for h = 1 and 2 since G is 2-connected.

Label Np(K1 u K2) : {zl, z2, ... zq} in the order on P from a to b. If q = 1 then {Cl, zl} is a cut set of G.

Thus q ~ 2. Call i < q good ff ]NKlvK,(Z~, Zi+l)] ~ 2, and call i bet ter ff either zi C Ne(K1) and Z~+l ~ Np(K2) or else Z~+l ~ Np(K1) and zi ~ Ne(Ks). From q ~ 2 follows the existence of a better i.

Case 1. There exist better integers j and 1. If zj C Np(K1) and Z~+l ~ Np(K~) choose Wl E NKI(z i) and w2 E NK~(z~I). In the other subcase choose Wl E NK,(Zi+O and w2 ~ NK~(zi). For h = 1 and 2 there exist longest wh,ch-paths Qh in H a n d ver- tices Vh E F(Kh) such that [Qhl --~ dH(vh) § 1, aS shown in the proof of Theorem 2.1. Set Q -- Q1 u Qo u Q2. Then IP[zi, Z~+l]] ~ 2 + tQI ~ 3 § dH(vl) + dH(v2). Similarly one can construct a path Q' in H such that ]P[zl, zt+l]l ~ 2 § [Q'[. Without loss of generality assume IQ'] ~ [Q[. Set t : = INe(vl) n Ne(vs)l.

If zi ~ Ne(K1) and zi+l E Ne(Ks) determine a path L = L[yl, y~] in H where Yl C NK,(Zi) and Ys C NK,(Z~+I). Then clearly [P[z~, z~+~]] ~ 2 + ILl ~ 5. Similarly IP[zi, zi+i]l ~ 5 if zi ~ Np(K2) and z~+~ ~/Ve(K1). Since there exist at least t -- 1 integers i such that IP[z~, z~+~]] ~ 5 one obtains [PHI ~ 2dp(v~) -~ 2de(vs) -- 2t -- 1 § 2(t -- 1) § 2([Q] - 3) hence

I P H l = 2 d ( v ~ ) § 2 4 7 where 8 ~ 0 . (5)

If z~ r N~(v~) ~ Ne(vs) for some i, there exist at least t integers i such that [P[z~, z~+i]l ~ 5 hence ~ ~ 2 in (5). Thus assume

Np(vl) = Np(vs) = N e ( K ~ Ks).

If z ~ N p ( H ) - Ne(K~ ~ K2), say z on P[z~, z~+~], determine v ~ Ng(Z). For h ---- 1 and 2 let Ra = Ra[va, v] be paths in H.

Then ]P[z~, zH]l = lP[z~, z]l § IP[z, z,§ -- 1 _~ 3 § [R d § IR2l ~ 7. If in particular z lies on P[zi, zi+~] determine for h ---- 1 and 2 paths Rn --- Ra[wa, v]

Qa in H. In this subcase [P[z1, z]l ~ 2 § IR~I and IP[z, z~+x]l ~ 2 + IR~I hence lP[z~, z~+~]l ~ 3 § IQ~I § IQsl _~ 5 + d~(vl) + d~(v~). Similarly [P[z. zt+~]l __~ 5 § dn(v~) § tin(v2) if z lies on P[z~, Zt+l]. One deduces tha t hrp(H) 4 = hre(K~ u K2) implies ~ ~ 2 in (5). Thus assume Np(H) = Ne(Vl) = Np(V2).

Case 1.1. q = 3.

Then {z~, %, %} is a cut set of G and d(v~) __. 4 for h = 1 and 2. If d(v~) ~ 5 then I P~] _~ d(v~) + 2d(v2) - - 2 by (5). Similarly IPHI ~2d(v~) § d(vs) -- 2 if d(v2) ~ 5.

136 H.A. Jung

Le t d(vl) : d(v2) = 4. I f ]Q] ~ 4 then [PH L ~ 3 ~- 2 ]Q] ~ 11 - - 3d(vh) - - 1 (h = ~, 2).

I f IQP - - 3 then cl ---- c2 and V(Kh) = {vh} for h ---- 1 and 2. B y the choice of Q0, K1 and K2 then H is a star, fu r ther IPHI _~ 9 = 2d(vh) ~- 1 and ]HI

d(vh) - - 1 (h = 1 and 2).

Case 1.2. q _~ 4.

Le t L be a longest vlv2.path in H. I f ILl _~ 5 then [P[z~, z~+l][ ~ 7 for all i < q hence ~ ~ 2 in (5). L e t ILl ~ 4. Then V(K1) = {vl} or V(K2) ---- {v2}, say V(K1) = {vl}. I f ]L[ _~ 4 t h e n rPHI ~ 4q ---- 4d(vi) - - 4.

Le t rL] ---- 3. Then cl ---- c2 and V(Kh) ---- {vh} for h ---- 1 and 2. B y construct ion, in t h a t subcase Cl is the only cut ve r t ex of H and ]K] ---- 1 for each componen t K of H - - c~. Hence H is a s ta r and

IP'] _~ 2d(vl) ~- 2d(v2) - - 7 ~ 2d(v~) + d(v2) -- 2.

Case 2. j is the only be t t e r integer.

For definiteness assume zj C Ne(K1) and zj+l E Np(K2). For h ---- 1 and 2 cons t ruc t Qa---Qa[wa, ca] and va E V(Ka) such t h a t IQa] ~ dR(va)-}-1 as in Case 1. Set Q - - Q1 u Q0 u Q2- B y const ruct ion wl E NK(ZS) and w2 ~ NK(zi+I).

Call a good integer i of t ype h if z~, z,+~ C Np(Ka) (h ~ 1, 2).

Case 2.1. There is a good integer I of type 1 wi th 1 ~ ] and a good integer m of t ype 2 with m ~ ].

Since ~ is the only be t t e r integer one has l ~ ] and ~" -~ 1 ~ m, hence Np(Ks) ___ {Zx . . . . , zi} and ~Vp(K2) ~ {zj§ . . . . . Zq}.

Determine dist inct vert ices az C NK~(zt) and bz E NK,(Zl+I) and a longest a,bz-path Rt in H .

We app ly Theorem 2.1 to the m a x i m a l subgraph K * of H spanned b y V(K1) u {cl}. I f V(K*) ~- V(Rt) set Yt : as. I f some componen t H* of K* -- R1 does no t contain cl pick y, ~ V(H*) such t h a t ]R~] ~ dK.(y,) + I ---- d,(y,) -}- 1. I f f inally K * - - R~ has only one componen t H * and H * contains c~ let y~ be an a r b i t r a r y ve r t ex in V(R~) --Nn~(H* ). I n a n y subcsse we have found some y~ ~ V(KI) such t h a t [R~I ~_ du(y~) -~- 1. Now [P[z~, z~] ] ~ 2 ~- [R~[ ~ 3 -~ dK(y,). Similar ly there are a p a t h R~ in H and y~ ( V ( K 2 ) such t h a t JP[zm, zm+~]] ~ 2 -~ IR~] __~ 3 ~- d~(ym).

Note t h a t IQ] -~ ]R,] ~- ]R~] ~ dH(vi) -~- all(v2) ~- d~(y,) + dH(ym) ~- 3. I f dH(V~) + d~(v2) ~ dH(y~) t - d(y,~) t hen I P ' [ ~ 2d~(y~) § 2d~(y~) - - 1 ~- [Q] § IR~] -}- ]R~] - - 3 and IP"] _~ 2d(y~) -~ 2d(ym) - - 1. Similar ly ]pH] ~ 2d(vi) -b 2d(v2) - - 1 if d~(y,) -}- d~(ym) ~ dn(v~) -}- dn(v~).

Case 2.2. There is a good integer I wi th 1 ~= j. Wi thou t loss of genera l i ty assume t h a t l is of t ype 1. I n view of Case 2.1 one also can assume t h a t there is no good integer m ~ ~ of t ype 2. Then N~(K~) ~ {z~ . . . . . zi+~} and Np(K~) ----- {Zj+l . . . . , zq} since j is the only be t t e r integer. Also NK.(zi+~ . . . . zq) = {w~} or ]-{- l ~ q .

Dete rmine R~ and y~ as in Case 2.1 so t h a t [P[z~, z~+l]] ~ 2 ~ [R~I ~ 3 -}- dK(y~). Then I P ' ] ~_ 2q - - 1 -}- ]R~I - - 1 + IQI - 1 yields IP'I ~dp(v~)

Longest paths joining given vertices in a graph 137

-]- dp(yl) + dp(v2) + dp(w~) -- 3 ~- dH(y~) • dH(Vl) -~ dH(V2) t h a t is IP'I ~ d(vl) + d(yl) + d(v2) ~- dp(w2) -- 3.

I f V(K2) ~ {w2} then {w2, c2} or {c2, zi+l} is a cut set. I f V(K2) =- {w2} then dp(w,) -= d(w2) - - 1 hence [pHI ~__ 4d(v) - - 4 for some v C V(H).

Case 2.3. There is no good integer i with i ~ j. Firs t assume 1 < j and j ~- 1 < q. Then _~p(K1) ~-- {Zl, ..., zj} and Np(K2) = {Zj+l, ..., Zq}. Also .NKI(P ) ---- {Wl} and _NKf(P ) =- {w2}. I f V(Ka) ~ {Wh} (h =- 1 or 2), then {ca, wh} is a cut set of G. Therefore V ( K 1 ) = {Wl} and V ( K 2 ) = {w2} hence ]P~l ~ 2dp(wl) ~- 2dp(w2) § 1 --~ 2d(wl) ~- 2d(w2) - - 3.

I f ~ = 1 and ~" + 1 < q then Np(K1) = {Zl} and {c I, zl} is a cut set of G. The subcase in which 1 < ~ and ~" ~ 1 -~ q is symmetric .

Final ly assume q - - 2 whence {cl, Zl, z2} is a cut set of G. Since {z 1, z2} is no t a cut set of G there is a ver tex z ~ Np(t t ) -- {zl, z2}. Choose v E N~(z). Since v r V(K1 u Ks) there exist for h =- 1 and 2 pa ths Ra =- Rh[wh, v] ~ Qa.

One obtains IP'[ ~-- [Q,I + ]Q2] + 3 hence [P~[ ~ dH(Vl) ~- dH(V2) -~- 5 ~= d(vl) + d(v2) + 1. Also ]HI ~_ d(va) for h ---- 1 and 2.

The proof of L e m m a 3.3 is complete.

Theorem 4 is a consequence of L e m m a 3.2 and L e m m a 3.3. I n view of L e m m a 3.1 and L e m m a 3.3, for the proof of Theorem 3 it remains to consider the case in which H is a s t ronghold in G, where G is k-connected and IHI

k - - 2 ~ 0. Abbreva t ingh : = IH[ a n d s : = [Np(g){ onehasd(v) = h + s - - 1 for v ~ V(t t) and ]pHI ~ (S -- 1) (h -{- 1) q- 1. Since (s - - 1) (h q- 1) ---- (k - - 1) • where b ---- (h - - k + 2) (s - - k) ~ 0 one deduces IP ~]

(k - - 1 ) (d(v) - - k + 2) + b + l w i t h b ~ k - - l i f s > k a n d h ~ 2 k - - 3 .

References

[1] J. A. Bo~DY, Longest paths and cycles in graphs of high degree, University of Waterloo Preprint CORR 80-76.

[2] J. A. BONDY and B. JACKSOn, Long paths between specified vertices of a block, to appear.

[3] H. ENO~mTO, Long paths and large cycles in finite graphs, Journal of Graph Theory 8 (1984), 287--301.

[4] P. ERDSS and T. GALLAI, On maximal paths and circuits in graphs, Acta Math. Acad. Sci. Hung. 10 (1959), 337--356.

[5] J. FOURNIER and P. FR~SSE, On a conjecture of Bondy, J. of Combinatorial Theory Ser. B 38 (1985).

[6] H. A. JuxG, Longest circuits in 3-connected graphs, Coll. Math. Soc. J. Bolyai, 37. Finite and infinite sets, Eger (1981), 403--438.

[7] S. C. LOCKE, Ph. D. Thesis, University of Waterloo (1981). [8] C. St. J. A. NAsH-WmLIAMS, Edge-disjoint hamiltonian circuits in graphs with

vertices of large valency, Studies in Pure Mathematics (pres. to R. Rado), Academic Press London (1971), 157--183.

Eingegangen am 2. 9. 1985, in revidierter Fassung am 24. 10. 1985

Anschrlft des Au~ors: Heinz Adols Jung, Faehbereich Mathematik der Technischen Universit~t, StraBe des 17. Juni 135, D - 1000 Berlin 12.