3-Coloring GraphsEmbedded in Surfaces

Yue ZhaoDEPARTMENT OF MATHEMATICS

UNIVERSITY OF CENTRAL FLORIDAORLANDO, FL 32816-1364

Received October 8, 1997; revised August 26, 1999

Abstract: In this article, we show that there exists an integer k(Σ) ≥ 5 suchthat, if G is a graph embedded in a surface Σ without i-circuits, 4 ≤ i ≤ k(Σ),then G is 3-colorable. c© 2000 John Wiley & Sons, Inc. J Graph Theory 33: 140–143, 2000

Keywords: vertex colorings; surfaces

1. INTRODUCTION

A graph G is said to be k-critical, if G has chromatic number k, but every propersubgraph of G is (k−1)-colorable. In 1975, Steinberg (see [2, 3, or 7]) conjecturedthat any 4-critical planar graph must contain a circuit of length 4 or 5. The followingrelaxation of Steinberg’s conjecture has been raised by Paul Erdos [5, 7].

Is there an integer k ≥ 5 such that, if G is a planar graph without i-circuits,4 ≤ i ≤ k, then G is 3-colorable?

In 1991, Abbott and Zhou [1] proved a theorem that implies that, if G is a planargraph without i-circuits, 4 ≤ i ≤ 11, then G is 3-colorable. In 1995, Borodin [4],Sanders, and Zhao [6] independently proved that, if G is a planar graph withouti-circuits, 4 ≤ i ≤ 9, then G is 3-colorable. In this article, we prove that, for everysurface Σ, there is an integer k(Σ) ≥ 5 such that, if G is embedded in Σ withouti-circuits, 4 ≤ i ≤ k(Σ), then G is 3-colorable.

In Section 2, we list several properties of 4-critical graphs. As an application ofthese properties, in Section 3, we prove the result for graphs embedded in a surfaceof characteristic ≤ 0.

Contract grant sponsor: National Science FoundationContract grant number: DMS 9896292

c© 2000 John Wiley & Sons, Inc.

2. SOME PROPERTIES OF 4-CRITICAL GRAPHS

In this section, we provide some general properties of 4-critical graphs embeddedin surfaces. To this end we need the following definitions.

A k-vertex, ≥ k-vertex is a vertex of degree k, at least k and a k-face, ≥ k-faceof an embedded graph is a face whose facial walk has length k, at least k. Letx ∈ V (G) ∪ F (G); we use d(x) to denote the degree of x if x is a vertex, or thesize of x if x is a face. Two faces are said to be adjacent, if they share a commonedge. Let f : v1 · · · vk be a k-face of an embedded graph G with k ≥ 4. We call fa triangle-alternating k-face, if (1) all vi, 1 ≤ i < k, are 3-vertices; (2) v2i−1, v2i

are both incident with a 3-face fi for i = 1, 2, . . . , bk2c. For the convenience of

the readers, throughout the rest of this article, whenever we talk about a triangle-alternating k-face f : v1 · · · vk, we always use ui to denote the vertex that is adjacentto both v2i−1 and v2i.

We omit the proofs of the following facts, since the proofs of Lemmas 1 and 2are similar to the proofs of Propositions 1 and 2 in [6], while the proof of Lemma3 is similar to the proof of Lemma 1.

Lemma 1. Let G be a 4-critical graph embedded in a surface Σ, then G does nothave a triangle-alternating 2k-face.

Lemma 2. Let G be a 4-critical graph embedded in a surface Σ. If G contains atriangle-alternating (2k + 1)-face (for k ≥ 2)f : v1 · · · v2k+1 such that d(v2k+1) =3, then none of ui is a 3-vertex.

Corollary 1. Let G be a 4-critical graph embedded in a surface Σ. If G con-tains a triangle-alternating (2k + 1)-face (for k ≥ 2)f : v1 · · · v2k+1, then eitherd(v2k+1) ≥ 4 or d(ui) ≥ 4 for all i = 1, . . . , k.

Lemma 3. Let G be a graph embedded in a surface Σ. Assume that G con-tains a (2k + 1)-face (for k ≥ 2)f : v1 · · · v2k+1 such that d(vi) = 3 for i =1, . . . , 2k, d(v2k+1) = 4, v2i, v2i+1 are incident with a 3-face fi: v2iv2i+1ui for1 ≤ i ≤ k, and v2k+1, v1 are incident with a 3-face fk+1: v2k+1v1uk+1. If eitherone of ui for i = 1, . . . , k − 1, is a 3-vertex or both uk and uk+1 are 3-vertices,then G is not 4-critical.

3. 3-COLORING GRAPHS EMBEDDED IN A SURFACE

Theorem 1. Any graph that can be embedded in a surface Σ of characteristicχ(Σ) ≤ 0 and has no i-circuits for 4 ≤ i ≤ k = 11 − 12χ(Σ) is 3-colorable.

Proof. Assume that our theorem is not true. Let G be a counterexample to ourtheorem with the minimum number of edges. Then G is 4-critical. By Euler’sFormula |V (G)| − |E(G)| + |F (G)| = χ(Σ), we have

∑

x∈V (G)∪F (G)

(4 − d(x)) = 4χ(Σ).

142 JOURNAL OF GRAPH THEORY

We call M(x) = 4−d(x) the initial charge of x. We reassign a new charge denotedby M ′(x) to each x ∈ V (G) ∪ F (G) according to the discharging rules below:

R1. For each 3-vertex v that is not incident with any 3-face, send 13 from v to each

face incident with it.R2. For each 3-vertex v that is incident with a 3-face and for each ≥ (k +1)-face

f incident with v, send 12 from v to f .

R3. For each 3-face f and for each ≥ (k + 1)-face f ′ adjacent to f , let v be avertex incident with both f and f ′, send 1

6 from f to f ′ via v.

Now we consider M ′(x) and show M ′(x) ≤ 0 for each x ∈ V (G) ∪ F (G).Let x be a 3-vertex. Since G has no i-circuits for 4 ≤ i ≤ k, x is either incident

with three ≥ (k + 1)-faces or two ≥ (k + 1)-faces and one 3-face. By R1, R2, xsends out 1. Hence, M ′(x) = 0.

Let x be a ≥ 4-vertex. Since x receives no charge by our discharging rules, wehave M ′(x) = M(x) ≤ 0.

Let x be a 3-face. Since x is adjacent to three ≥ (k + 1)-faces, by R3, x sendsout 1 via vertices incident with it. Hence, M ′(x) = 0.

Let x be a 2j-face with 2j ≥ (k + 1). By Lemma 1, x is incident with at most(2j − 2) 3-vertices, which are incident with 3-faces. By R1, R3, x receives 1

3 fromeither a 3-vertex that is incident with it and is not incident with any 3-face, or two3-faces (adjacent to x) via a ≥ 4-vertex that is incident with both x and the 3-faces.By R2, x receives 1

2 from each 3-vertex that is incident with both x and a 3-face. ByR3, x receives 1

6 from a 3-face (adjacent to x) via a vertex that is incident with bothx and the 3-face. Hence, we have M ′(x) ≤ M(x) + 2

3 + 12(2j − 2) + 1

6(2j − 2) =10−2j

3 ≤ 10−k−13 = 9−11+12χ(Σ)

3 = 4χ(Σ) − 23 .

Let x be a (2j + 1)-face for (2j + 1) ≥ (k + 1). Then x is incident with atmost 2j 3-vertices, which are incident with 3-faces. By R1, R2, and R3, we haveM ′(x) ≤ M(x) + 1

3 + 2j2 + 2j

6 = 10−2j3 ≤ 10−k

3 = 10−11+12χ(Σ)3 = 4χ(Σ) − 1

3 .Hence, we have

4χ(Σ) =∑

x∈V (G)∪F (G)

M(x) =∑

x∈V (G)∪F (G)

M ′(x) ≤ 4χ(Σ) − 13,

a contradiction. Thus, our theorem is true.

The result in Theorem 1 might be far from the best possible. In fact, in the proof ofTheorem 1, we used only one face f with d(f) ≥ k +1 to obtain the contradiction.Since Theorem 1 assures us of the existence of k(Σ), the question now is what isthe smallest integer k(Σ) that guarantees any graph that is embedded in a surfaceΣ and has no i-circuits for 4 ≤ i ≤ k(Σ) to be 3-colorable.

By using a refinement of the above method and mimicking the proof in [6], onecan obtain the following result that deals with the particular case of a surface ofcharacteristic χ ≥ 0 and has a better bound in the case of χ = 0.

Theorem 2. Any graph that can be embedded in a surface of characteristic χ ≥ 0and has no i-circuits for 4 ≤ i ≤ 9 is 3-colorable.

3-COLORING GRAPHS EMBEDDED IN SURFACES 143

ACKNOWLEDGMENTS

The author wishes to thank the referees for their many valuable comments, whichled to the present form of this article.

References

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[3] V. A. Aksionov and L. S. Melnikov, Some counterexamples associated withthe three color problem, J Comb Theory Ser B28 (1980), 1–9.

[4] O. V. Borodin, Structural properties of plane graphs without adjacent trianglesand an application to 3-colorings, J Graph Theory 21 (1996), 183–186.

[5] P. Erdos, Informal discussion during the conference, Quo Vadis, Graph The-ory? Univ Alaska, Fairbanks, August, 1990.

[6] D. P. Sanders and Y. Zhao, A note on the three color problem, Graph Combin11 (1995), 91–94.

[7] R. Steinberg, The state of the three color problem, Ann Discrete Math 55(1993), 211–248.