on light edges and triangles in projective planar graphs

On Light Edges and Triangles in Projective Planar Graphs

Daniel P. Sanders* DEPARTMENT OF MATHEMATKS

THE OHlO STATE UNlVERSlN COLUMBUS, OHIO 432 I0

e-mail: dsanders@math. ohio-state. edu

ABSTRACT

An edge or face of an embedded graph is light if the sum of the degrees of the vertices incident with it is small. This paper parallelizes four inequalities on the number of light edges and light triangles from the plane to the projective plane. Each of the four inequalities is shown to be the best possible. 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

Let an i-vertex be a vertex of degree i. Let an i, j-edge be an edge joining an i-vertex to a j-vertex. Let an i, j, k-face be a face bounded by a triangle containing an i-vertex, a j-vertex, and a k-vertex. Given a plane graph G, let ui (or ui(G) if needed for distinction) be the number of i-vertices of G, let e , , , be the number of i , j-edges of G, and let f i , j , k be the number of i , j, k-faces of G. Let an edge of a graph G be light if it is an i, j-edge such that i + j 5 11, or i = 3 and i + j 5 13. Let a triangle of a graph G be light if it is an i, j , k-face such that i + j + k I 17.

Kotzig [8] showed that every 3-connected planar graph contains an i, j-edge, where i + j I 13. This result is best possible, in that the 13 cannot be replaced by a smaller number. It is not best possible, in another sense. Grunbaum [12] conjectured an inequality showing that there are a large number of these type edges in 3-connected planar graphs. After some progress in this area by Jucovic [7], the best possible inequality of this type was found by Borodin [2] for the broader class of normal planar maps, or plane graphs such that each vertex, as well as each face, is incident with at least three edges. In the same paper [2], he established a parallel result for normal maps of minimum degree four.

*This research was supported by the Office of Naval Research, Grant N00014-92-J- 1965.

Journal of Graph Theory, Vol. 21, No. 3, 335-342 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0364-9024/96/030335-08

336 JOURNAL OF GRAPH THEORY

Lemma 1 (Borodin). Every normal planar map satisfies the following:

and each of these coefficients is best possible.

Lemma 2 (Borodin). Every normal planar map of minimum degree four satisfies the following:


Wernicke [13] had established the existence of light edges in planar graphs of minimum degree five in his work on the Four Color Problem. After contributions by Grunbaum [5], Fisk [6], Grunbaum and Shephard [6], and Borodin [3], the best possible light edge inequality for planar graphs of minimum degree five was found by Borodin and Sanders [4]. Note that the best possible coefficient on e5.5 is not 8/15, like in Lemmas 1 and 2.

Proposition (Borodin, Sanders). the following:

Every normal planar map of minimum degree five satisfies


Not only do planar graphs of minimum degree five contain light edges, they also contain light triangles. This is not true for planar graphs with minimum degree less than five, by the example of a circuit with a vertex put in each face, each joined to each vertex of the circuit. The existence of light triangles in plane triangulations of minimum degree five was conjectured by Kotzig [9]. Borodin [ l ] proved that all planar graphs of minimum degree five contain light triangles, and then in 131 proved an inequality showing that they contain a number of light triangles. Borodin and Sanders [3,4] showed that each coefficient of the inequality is best possible.

Lemma 3. Every planar graph of minimum degree five satisfies the following:


The existence of edges of small weight (the sum of the ends of the edge) has been shown for orientable surfaces other than the sphere. For example, Grunbaum and Shephard [6] showed that every toroidal triangulation contains an edge of weight at most 15. This paper shows that not only do projective planar graphs of minimum degree three contain light edges, but they satisfy inequalities extremely similar to those for planar graphs. The similarity is parallel to that of minor vertices (vertices of degree at most five). Euler’s formula gives (with some terms left out) 3u3 + 2u4 + u5 I 12 for the plane and 3113 + 2u4 + U S 2 6 for the projective plane. The only inequality which is not completely parallel is the light edge inequality for

LIGHT EDGES AND TRIANGLES 337

FIGURE 1.

graphs of minimum degree five. For the plane, the coefficient of e5,5 went from 8/15 to 7/15. For the projective plane, it is lowered to 16/35. All the other coefficients are identical to the coefficients in the planar inequalities.

Since the coefficients are identical, most of the proofs are simple, from the following fact: For every projective planar graph G , there is a planar graph H where, for each i, j , and k, the following are true: u i ( H ) = 2ui(C), e i , j ( H ) = 2ei,,(G), and f i , j , k ( H ) = 2fi , j ,k(G). Let such a graph H be called a planar cover of the projective planar graph G . This simple fact is stated as Lemma 4.

Lemma 4. Every projective planar graph has a planar cover.

An embedding of the planar cover can be obtained from the projective planar embedding, for example, by embedding a copy of the projective plane into each of the Northern and Southern hemispheres of a sphere [see 10, 111. Lemma 4 proves the existence of the inequalities. Then to show that the coefficients are best possible, projective planar graphs must be demonstrated which satisfy the inequalities with equality.

As mentioned before, this technique will not yield the best possible light edge inequality for projective planar graphs of minimum degree five (Theorem 3 below). Lemmas 1 to 3 as well as the Proposition were each originally proved by means of the discharging method [see, 2, 3, 41. It seems that a similar set of discharging rules could yield a proof for Theorem 3, but they would be complicated. Instead, a simpler counting argument works. Let a half-edge be an ordered pair (e , u), where e is an edge, and v is a vertex incident with e . Let a half-edge


( e , u ) be halflight if the degree of v is less than six. Let a half-edge ( e , u ) be light if e is a light edge. These terms will be used in the proof of Theorem 3.

2. THE PROJECTIVE LIGHT INEQUALITIES

Theorem 1. Every normal projective planar map satisfies the following:


Proof. To see that the coefficient of e3.8 is best possible, consider G6 in Figure 2, which has

e3,8 = 12, and no other light edges. To see that the coefficient of e3.10 is best possible, consider G8 in Figure 2, which has e3,lo = 30, and no other light edges. To see that the coefficient of e4,6 is best possible, consider G11 in Figure 3, which has e4,6 = 12, and no other light edges. To see that the coefficient of e4.7 is best possible, consider G12 in Figure 3, which has e4.7 = 36, and no other light edges. To see that the coefficient of e5.6 is best possible, consider G14 in Figure 4, which has e5.6 = 30, and no other light edges.

The inequality follows directly from Lemmas 1 and 4.

FIGURE 2


G11 G12

FIGURE 3.

To see that the coefficient of e3.3 is best possible, consider GI in Figure 1, which has e3.3 = 1, e3 ,8 = 4, and no other light edges. To see that the coefficient of e3.4 is best possible, consider G2 in Figure 2, which has e3.4 = 1 , e3,8 = 7, and no other light edges. To see that the coefficient of e3,7 is best possible, consider G5 in Figure 2, which has e 3 , ~ = 3, e3.8 = 6, e4,7 = 6, and no other light edges. To see that the coefficient of e3,9 is best possible, consider G7 in Figure 2, which has e3.9 = 12, e3.10 = 15, and no other light edges. To see that the coefficient of e4.4 is best possible, consider G9 in Figure 3, which has e4.4 = 1, e4.6 = 6, e4,7 = 8, and no other light edges. To see that the coefficient of e4.5 is best possible, consider Glo in Figure 3, which has e 4 , ~ = I, e4.6 = 4, e4,7 = 15, e5,6 = 2, and no other light edges. To see that the coefficient of e5.5 is best possible, consider G13 in Figure 4, which has e5.5 = I , e4.6 = 8, e4,7 = 4, e5 .6 = 4, and no other light edges.

To see that the coefficient of e3,5 is best possible, consider G3 in Figure 1, which has e3.5 =

1, e3.7 = 2, e4.6 = 4, e4.7 = 4, e5,6 = 2, and no other light edges. To see that the coefficient of e3.6 is best possible, consider G4 in Figure 1 , which has e3.6 = 2, e3.7 = 4, e4,7 = 8, and no other light edges. I Theorem 2. Every normal projective plane map of minimum degree four satisfies the following:

and each of these coefficients is, best possible.


G l 5

FIGURE 4.

Proof. That each of the coefficients is best possible follows from the graphs G9, Glo, G11, (312, G13,


and G14, which were described in the proof of Theorem 1.

Theorem 3. Every projective planar graph of minimum degree five satisfies the following:


Let G be a projective plane graph such that the minimum degree of G is five. Without loss of generality, (by adding edges) G is a triangulation. Since G is projective planar, it satisfies 1vEV(G) (6 - deg(v)) = 6, or Ex=5(6 - x)ux = 6. The number h of half- light half-edges in G is equal to 5115, or 5(6 + I x r 7 ( x - 6 ) ~ ~ ) . Let g be the number of light half-edges in G .

If g 2 30, then the inequality above is satisfied. Thus the only cases which need to be considered are those which have g < 30. From the sum above, for each vertex v of degree x > 6, there are an additional 5x - 30 half-light half-edges, and (just because u is adjacent to at most x 5-vertices) at most an additional x half-light half-edges which are not light. Thus there are a number of 7-vertices which are adjacent to more than five 5-vertices

If g = 29, then there is a 7-vertex adjacent to at least six 5-vertices. This implies that e5,s 1 5. Thus the inequality is satisfied here.

Prooj


If g = 28, then there are two cases. If there is a 7-vertex adjacent to seven 5-vertices, then e5,5 2 7. If there are two 7-vertices each adjacent to six 5-vertices, then e5,5 2 9. In either case, the inequality is satisfied.

Finally, consider the case when g 5 27. 'Avo cases are also possible here. First, there could be a 7-vertex adjacent to seven 5-vertices, and another 7-vertex adjacent to at least six 5- vertices. Second, there could be three 7-vertices each adjacent to six 5-vertices. In each case, e5.5 2 10, implying the inequality.

To see that the coefficient of e5,6 is best possible consider G14 in Figure 4, which has e5.5 = 0 and e5.6 = 30. To see that the coefficient of e5.5 is best possible, consider G15 in Figure 4, which has e5,5 = 7 and e5,6 = 14.

Theorem 4. Every projective planar graph of minimum degree five satisfies the following:


Proof. To see that the coefficient of f5,5,5 is best possible, consider H I in Figure 5, which has

f 5 . 5 , ~ = 4, and no other light triangles. To see that the coefficient of f5.5.6 is best possible, consider H2 in Figure 5 , which has f5,5,6 = 8, and no other light triangles. To see that the coefficient of f5,6,6 is best possible, consider H4 in Figure 5, which has f5.6.6 = 18, and no


*2

*3

FIGURE 5.


other light triangles. Finally, to see that the coefficient of f 5 , 5 , 7 is best possible, consider H3

in Figure 5, which has f 5 . 5 , ~ = 8, f5.6.6 = 8, and no other light triangles. I

References

[l] O.V. Borodin, Solution of problems of Kotzig and Grunbaum concerning the isolation of cycles in planar graphs, Math Notes 46 (1989), 835-837 (English translation), Mat. Zumetki 46 ( 5 ) (1989), 9-12. (Russian).

[2] O.V. Borodin, Precise lower bound for the number of edges of minor weight in planar maps, Math. Slov. 42 (1992), 129-142.

[3] 0. V. Borodin, Structural properties of planar maps with the minimal degree 5 , Math. Nachr., 158 (1992), 109-117.

[4] O.V. Borodin and D.P. Sanders, On light edges and triangles in planar graphs of minimum degree five, Math. Nachr., 170 (1994). 19-24.

[5] B. Grunbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973), 390-408. [6] B. Grunbaum and G. C. Shephard, Analogues for tilings of Kotzig’s theorem on minimal

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[S] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat. Cus. 5 (1955), 101-1 13

[9] A. Kotzig, From the theory of Euler’s polyhedrons, Mat. Cus. 13 (1963), 20-34 (Russian). [lo] S. Negami, Graphs which have no finite planar covering, Bull. Inst. Math. Acud. Sinica

[ 111 S. Negami, The spherical genus and virtually planar graphs, Discrete Math. 70 (1988),

[ 121 Teoria Combinatoria, Proc. Intern. Colloq. Rome 1973, Accademia nacionale dei lincei,

[13] P. Wernicke, Uber den kartographischen Vierfarbensatz, Math. Ann. 58 (1904), 413-426.

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Received August 26, 1994

on light edges and triangles in projective planar graphs

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