Total colorings of planar graphs with small maximum degree

Advertisement


Advertisement
Similar documents
M-Degrees of Quadrangle-Free Planar Graphs

An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph

Labeling outerplanar graphs with maximum degree three

On the k-path cover problem for cacti

Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR

Mean Ramsey-Turán numbers

Cycles in a Graph Whose Lengths Differ by One or Two

On Integer Additive Set-Indexers of Graphs

On the maximum average degree and the incidence chromatic number of a graph

Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Every tree contains a large induced subgraph with all degrees odd

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

Odd induced subgraphs in graphs of maximum degree three

BOUNDARY EDGE DOMINATION IN GRAPHS

A 2-factor in which each cycle has long length in claw-free graphs

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Graphs without proper subgraphs of minimum degree 3 and short cycles

Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

All trees contain a large induced subgraph having all degrees 1 (mod k)

The Independence Number in Graphs of Maximum Degree Three

Cycles and clique-minors in expanders

Finding and counting given length cycles

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

Product irregularity strength of certain graphs

Non-Separable Detachments of Graphs

On the independence number of graphs with maximum degree 3

Best Monotone Degree Bounds for Various Graph Parameters

Divisor graphs have arbitrary order and size

CS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin

V. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005

Extremal Wiener Index of Trees with All Degrees Odd

Clique coloring B 1 -EPG graphs

On parsimonious edge-colouring of graphs with maximum degree three

P. Jeyanthi and N. Angel Benseera

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

Special Classes of Divisor Cordial Graphs

Single machine parallel batch scheduling with unbounded capacity

3. Eulerian and Hamiltonian Graphs

arxiv: v2 [math.co] 30 Nov 2015

The degree, size and chromatic index of a uniform hypergraph

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

GRAPH THEORY LECTURE 4: TREES

The positive minimum degree game on sparse graphs

When is a graph planar?

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE

A simple criterion on degree sequences of graphs

Degree-associated reconstruction parameters of complete multipartite graphs and their complements

Discrete Mathematics Problems

Social Media Mining. Graph Essentials

On one-factorizations of replacement products

8. Matchings and Factors

Coloring a set of touching strings

Linear Algebra and its Applications

SCORE SETS IN ORIENTED GRAPHS

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

Minimum degree condition forcing complete graph immersion

arxiv: v5 [math.co] 26 Dec 2014

The chromatic spectrum of mixed hypergraphs

The Open University s repository of research publications and other research outputs

Triangle deletion. Ernie Croot. February 3, 2010

Embedding nearly-spanning bounded degree trees

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

High degree graphs contain large-star factors

The minimum number of distinct areas of triangles determined by a set of n points in the plane

A Turán Type Problem Concerning the Powers of the Degrees of a Graph

LIGHT SUBGRAPHS IN PLANAR GRAPHS OF MINIMUM DEGREE 4 AND EDGE-DEGREE 9

List Edge and List Total Colourings of Multigraphs

FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem

most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia

Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

On end degrees and infinite cycles in locally finite graphs

Removing Even Crossings

YILUN SHANG. e λi. i=1

Large induced subgraphs with all degrees odd

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

8.1 Min Degree Spanning Tree

Exponential time algorithms for graph coloring

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Analysis of Algorithms, I

On-line Ramsey Theory for Bounded Degree Graphs

Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

DEGREE-ASSOCIATED RECONSTRUCTION PARAMETERS OF COMPLETE MULTIPARTITE GRAPHS AND THEIR COMPLEMENTS

Examination paper for MA0301 Elementær diskret matematikk

Strong Ramsey Games: Drawing on an infinite board

Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, Chapter 7: Digraphs

arxiv: v1 [math.co] 7 Mar 2012

Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Cycle transversals in bounded degree graphs

Degree Hypergroupoids Associated with Hypergraphs

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

On an anti-ramsey type result

Generalized Induced Factor Problems

Removing even crossings

Conductance, the Normalized Laplacian, and Cheeger s Inequality

Network (Tree) Topology Inference Based on Prüfer Sequence

Advertisement
Transcription:

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong University, Jinan, 50100, P.R. China Zaozhuang No. middle school, Shandong, 77100, P.R. China Abstract Let G be a planar graph of maximum degree Δ and girth g, and there is an integer t(> g) such that G has no cycles of length from g + 1 to t. Then the total chromatic number of G is Δ + 1 if (Δ, g, t) {(5, 4, 6), (4, 4, 17)}; or Δ = and (g, t) {(5, 1), (6, 11), (7, 11), (8, 10), (9, 10)}, where each vertex is incident with at most one gcycle. Keywords: Total coloring; Planar graph; Cycle; Girth MSC: 05C15 1 Introduction All graphs considered in this paper are simple, finite and undirected, and we follow [6] for the terminologies and notations not defined here. Let G be a graph. We use V (G), E(G), Δ(G) and δ(g) (or simply V, E, Δ and δ) to denote the vertex set, the edge set, the maximum degree and the minimum degree of G, respectively. For a vertex v V, let N(v) denote the set of vertices adjacent to v, and let d(v) = N(v) denote the degree of v. A k-vertex, a k + -vertex or a k -vertex is a vertex of degree k, at least k or at most k respectively. A k-cycle is a cycle of length k, and a -cycle is usually called a triangle. A total-k-coloring of a graph G is a coloring of V E using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ (G) of G is the smallest integer k such that G has a total-k-coloring. Clearly, χ (G) Δ + 1. This work is supported by a research grant NSFC ((109711) of China. Corresponding author.e-mail addresses: jlwu@sdu.edu.cn. 1

Behzad [1] posed independently the following famous conjecture, which is known as the Total Coloring Conjecture (TCC). Conjecture.For any graph G, Δ + 1 χ (G) Δ +. This conjecture was confirmed for a general graph with Δ 5. But for planar graph, the only open case is Δ = 6 (see [9],[]). Interestingly, planar graphs with high maximum degree allow a stronger assertion,that is, every planar graph with high maximum degree Δ is (Δ + 1)-totally-colorable. This result was first established in [] for Δ 14, which was extended to Δ 9 (see [10]). For 4 Δ 8, it is not known whether that the assertion still holds true. But there are many related results by adding girth restrictions, see [7, 8, 11, 1, 14, 15, 16]. We present our new results in this paper. Theorem 1. Let G be a planar graph of maximum degree Δ and girth g, and there is an integer t(> g) such that G has no cycles of length from g + 1 to t. Then the total chromatic number of G is Δ + 1 if (a) (Δ, g, t) = (5, 4, 6) or (b)(δ, g, t) = (4, 4, 17). Borodin et al. [5] obtained that if a planar graph G of maximum degree three contains no cycles of length from to 9, then χ (G) = Δ+1. In the following, we make further efforts on the total-colorability of planar graph on the condition that G contains some k-cycle,where k (5,, 9). We get the following result. Theorem. Let G be a planar graph of maximum degree and girth g, each vertex is incident with at most one g-cycle and there is an integer t(> g) such that G has no cycles of length from g + 1 to t. Then χ (G) = Δ + 1 if one of the following conditions holds. (a) g = 5 and t 1, (b) g = 6 and t 11, (c) g = 7 and t 11; (d) g = 8 and t 10, (e) g = 9 and t 10. We will introduce some more notations and definitions here for convenience. Let G = (V, E, F ) be a planar graph, where F is the face set of G. The degree of a face f, denoted by, is the number of edges incident with it, where each cut-edge is counted twice. A k-face or a k + -face is a face of degree k or at least k, respectively. Let n k (v) be the number of k-vertices adjacent to v and n k (f) be the number of k-vertices incident with f. Proof of Theorem 1 Let G be a minimal counterexample to Theorem 1 in terms of the number of vertices and edges. Then every proper subgraph of G is (Δ + 1)-totally-colorable. Firstly, we investi-

gate some structural properties of G,which will be used to derive the desired contradiction completing our proof. Lemma. G is -connected and hence, it has no vertices of degree 1 and the boundary b(f) of each face f in G is exactly a cycle (i.e. b(f) cannot pass through a vertex v more than once). Lemma 4. [4] G contains no edge uv with min{d(u), d(v)} Δ and d(u) + d(v) Δ + 1. Lemma 5. [] The subgraph of G induced by all edges joining -vertices to Δ -vertices is a forest. Lemma 6. [5] If Δ 5, then no -vertex is adjacent two -vertices. Let G be the subgraph induced by all edges incident with -vertices of G. Then G is a forest by Lemma 5. We root it at a 5-vertex. In this case, every -vertex has exactly one parent and exactly one child, which are 5-vertices. Since G is a planar graph, by Euler s formula, we have v V (d(v) 4) + f F ( 4) = 8 < 0. Now we define the initial charge function ch(x) of x V F to be ch(v) = d(v) 4 if v V and ch(f) = 4 if f F. It follows that x V F ch(x) < 0. Note that any discharging procedure preserves the total charge of G. If we can define suitable discharging rules to change the initial charge function ch to the final charge function ch on V F, such that ch (x) 0 for all x V F, then we get an obvious contradiction. Now we design appropriate discharging rules and redistribute weights accordingly. For a vertex v, we define f k (v) or f + k (v) to be the number of k-faces or k+ -faces incident with v, respectively. To prove (a), our discharging rules are defined as follows. R11. Each -vertex receives from its child. R. Each -vertex v receives 1 f + 7 (v) from each of its incident 7+ -faces. R1. Each 5 + -vertex receives 1 from each of its incident 7+ -faces. Next,we will check ch (x) 0 for all x V F. Let f F (G). If = 4, then ch (f) = ch(f) = 0. Suppose = 7. Then n (f) 4 by Lemma 6. Moreover, every 7 + -face sends at most 1 to its incident -vertices by R and 1 to its incident 5-vertices by R1. So we

have ch (f) ch(f) 4 1 1 = 0. Suppose 8. Then n (v) by Lemma 4 and Lemma 6. Thus, ch (f) ch(f) ( 1 ) ( ) 1 5 8 0 9 by R and R1. Let v V (G). If d(v) =, then ch (v) = ch(v)+ = 0. If d(v) =, then f + 7 (v) and it follows from R that ch (v) = ch(v)+f 7 + (v) 1 = 0. If d(v) = 4, then f + 7 (v) ch (v) = ch(v) = 0. If d(v) = 5, then f 7 + (v). Moreover, it may be the child of at most one -vertex. Thus ch (v) ch(v) + 1 = 0 by R1. Suppose d(v) 6. Then v is incident with at most d(v) 4-faces and it may be the the parent of at most one -vertex. So ch (v) ch(v) + (d(v) d(v) ) 1 = 7d(v) 6 > 0. 6 Note that (a) implies that (b) is true if Δ 5. So it suffice to prove (b) by assuming Δ = 4. Since G is a planar graph, by Euler s formula, we have ( 6) = < 0. v V (d(v) 6) + f F Now we define the initial charge function ch(x) of x V F to be ch(v) = d(v) 6 if v V and ch(f) = 6 if f F. It follows that x V F ch(x) < 0. To prove (b), we construct the new charge ch (x) on G as follows. R1 Each ( 18)-face gives 1 6 to its incident vertices. R Each -vertex gets from its child and 1 from its parent. R Let f be a 4-face. If f is incident with a -vertex, then it gets from each of its incident + -vertices. If f is incident with no -vertices, then it gets 1 from each of its incident vertices. The rest of this paper is devoted to checking ch (x) 0 for all x V F. Let f F (G). If = 4, then ch (f) = ch(f) + max{, 1 4} = 0. If 18, then ch (f) = ch(f) r (1 6 ) = 0 by R1. r Let v V (G). If d(v) =, then ch (v) = ch(v) + + 1 = 0 by R. If d(v) =, then f 18(v) + and f 4 (v) 1,and it follows from R1 and R that ch (v) = ch(v)+ > 0. Suppose that d(v) = 4. Then ch(v) = 4 6 =. If n (v) 1, then v sends at most n (v)+ to all its adjacent -vertices by R. If n (v) 4, then f 4 (v) 1 by Lemma 5, and it follows that ch (v) ch(v) n (v)+ + = 14 n (v) > 0 by R1 and R. If 6 1 n (v), then f 4 (v), and it follows that ch (v) ch(v) n (v)+ + = n (v) 0. If n (v) = 0, we have f 4 (v). Moreover, each 4-face incident with v contains no -vertices. By R, we have ch (v) ch(v) + 1 > 0. Now we complete the proof of Theorem 1. 4

Proof of Theorem A (k) vertex is a -vertex adjacent to exactly k -vertices. Let G be a minimal counterexample to Theorem in terms of the number of vertices and edges. By minimality of G, it has the following result. Lemma 7. [5] (a) no -vertex is adjacent to two -vertices; (b) no -vertex is adjacent to a -vertex and a ()-vertex ; (c) no -vertex is adjacent to three -vertices. Let G be the bipartite subgraph of G comprising V and all edges of G that join a - vertex to a -vertex. Then G has no isolated -vertices by Lemma 7(a), and the maximum degree is at most by Lemma 7(c), and any component of G is a path with more than one edges must end in two -vertices by Lemma 7(b). It follows that n n. So we can find a matching M in G saturating all -vertices. If uv M and d(u) =,v is called the -master of u. Each -vertex has one -master and each vertex of degree Δ can be the -master of at most one -vertex. Since G is a planar graph, by Euler s formula, we have v V (d(v) 6) + f F ( 6) = < 0. Now we define the initial charge function ch(x) of x V F to be ch(v) = d(v) 6 if v V and ch(f) = 6 if f F. It follows that x V F ch(x) < 0. Note that any discharging procedure preserves the total charge of G. If we can define suitable discharging rules to change the initial charge function ch to the final charge function ch on V F, such that ch (x) 0 for all x V F, then we get an obvious contradiction. Now we design appropriate discharging rules and redistribute weights accordingly. R1 Each ( 5)-face gives 6 to its incident vertices. R Each -vertex receives 6 from its -master. t+1 g Let ch (x) be the new charge obtained by the above rules for all x V F. If f F (G), then ch (f) = ch(f) 6 = 0 by R1. Let v V (G). Suppose d(v) =. Then v can be the -master of at most one -vertex, and v sends at most 6 to -vertex t+1 g by R. In addition, If v is incident with a g-face, then the other faces incident with v are two (t + 1) + -faces, for G has no cycles of length from g + 1 to t. Thus, v receives ( 6 g ) from its incident g-face and ( 6 ) from each of its incident (t + t+1 1)+ -face by R1. So ch (v) ch(v) + ( 6 ) + ( 6 ) ( 6 ) = 0 for all g and t. Otherwise, v t+1 g t+1 g 5

is incident with three (t + 1) + -faces, then ch (v) ch(v) + ( 6 ) ( 6) = t+1 t+1 g 6 6 > 0, for t + 1 > g. Suppose d(v) =. Then v receives at most 6 g t+1 t+1 g from its -master by R1. If v is incident with a g-face, since G has no cycles of length from g + 1 to t, then the other face incident with v is a (t + 1) + -face, and it follows that ch (v) ch(v)+( 6 )+( 6 )+( 6 ) = 0 for all g and t. Otherwise, v is incident t+1 g t+1 g with two (t + 1) + -faces, then ch (v) ch(v) + ( 6 ) + ( 6) = 4 6 > 0. t+1 t+1 g t+1 g From the above, we can see that ch (f) = ch(f) 6 Suppose d(v) =.So ch (v) ch(v) + ( 6 ) + ( 6 ) ( 6 t+1 g t+1 g When v is incident with three (t+1) + -faces, then ch (v) ch(v)+( 6 6 6 g = 0 for all f F (G). ) = 0 for all g and t. ) ( 6) = t+1 t+1 g > 0, for t+1 > g. Suppose d(v) =. If v is incident with a g-face and a t+1 (t+1)+ -face, then ch (v) ch(v) + ( 6 t+1 ) + ( 6 g ) + ( t+1 6 g ) = 0 for all g and t.when v is incident with two (t + 1) + -faces, then ch (v) ch(v) + ( 6 ) + ( 6) = 4 6. So t+1 t+1 g t+1 g when g = 5, then t 1;when g = 6, then t 11;when g = 7, then t 11; when g = 8, then t 10; when g = 9, then t 10, and it follows that ch (v) 0. Our proof of Theorem is now complete. References [1] M. Behzad, Graphs and their chromatic numbers, Ph. D. Thesis, Michigan State University, 1965. [] O. V. Borodin, On the total coloring of planar graphs, J. Reine Angew. Math. 94 (1989) 180-185. [] O.V. Borodin, A.V. Kostochka and D.R. Woodall, List edge and list total colourings of multigraphs, J. Combin. Theory Ser. B 71 (1997) 184-04. [4] O. V. Borodin, A.V. Kostochka and D.R. Woodall, Total colorings of planar graphs with large maximum degree, J. Graph Theory 6 (1997) 5-59. [5] O. V. Borodin, A.V. Kostochka and D.R. Woodall, Total colourings of planar graphs with large girth, Europ. J. Combin. 19 (1998) 19-4. [6] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, MacMillan, London, 1976. 6

[7] D. Du, L. Shen and Y. Wang, Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable, Discrete Applied Math. 157 (009) 605-604. [8] J. F. Hou, Y. Zhu, G. Z. Liu, J. L. Wu and M. Lan, Total Colorings of Planar Graphs without Small Cycles, Graphs and Combinatorics 4( 008) 91-100. [9] A. V. Kostochka, The total chromatic number of any multigraph with maximum degree five is at most seven, Discrete Math. 16 (1996) 199-14. [10] L. Kowalik, J-Sébastien.Sereni and R. Skrekovski, Total colorings of planar graphs with maximum degree nine, SIAM J. Discrete Math. (008)146-1479. [11] Q. Ma, J. L. Wu and X. Yu, Planar graphs without 5-cycles or without 6-cycles, Discrete Mathematics 09:10( 009) 998-005. [] D. P. Sanders and Y. Zhao, On total 9-coloring planar graphs of maximum degree seven, J. Graph Theory 1 (1999) 67-7. [1] P. Wang and J. L. Wu, A note on total colorings of planar graphs without 4-cycles. Discuss. Math. Graph Theory 4 (004) 5-15. [14] J.B. Li, G.Z.Liu. On f-edge cover coloring of nearly bipartite graphs. Bull. Malays. Math. Sci. Soc. () 4 (011), no., 47 5. [15] A.J. Dong, X. Zhang and G.J. Li. Equitable Coloring and Equitable Choosability of Planar Graphs without 5- and 7-Cycles. Bull. Malays. Math. Sci. Soc. (). Accepted. [16] H.Yu. Chen, X.Tan and J.L. Wu. The Linear Arboricity of Planar Graphs without 5-cycles with Chords. Bull. Malays. Math. Sci. Soc. (). Accepted. 7