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Enqiang Zhu    Chanjuan Liu    Yongsheng Rao Corresponding author: ysrao2018@163.com
Abstract

A total -coloring of a graph is an assignment of colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph has a total ()-coloring, where is the maximum degree of . This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph of or with some restrictions has a total -coloring. In particular, in [Shen and Wang, “On the 7 total colorability of planar graphs with maximum degree 6 and without 4-cycles”, Graphs and Combinatorics, 25: 401-407, 2009], the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent 4-cycles or not incident with three cycles of size for some .

On a Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable]On a Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

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    Enqiang Zhu and Chanjuan Liu and Yongsheng Raothanks: Corresponding author: ysrao2018@163.com

    Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China

    School of Computer Science and Technology, Dalian University of Technology, Dalian, China

     

1 Introduction

Throughout the paper, we consider only simple, finite and undirected planar graphs, and follow Bondy and Murty (2008) for terminologies and notations not defined here. Given a graph , we use and to denote the vertex set and the edge set of , respectively. For a vertex , we denote by the degree of in and let . A -vertex, -vertex or -vertex is a vertex of degree , at most or at least . For a planar graph , we always assume that is embedded in the plane, and denote by the set of faces of . The degree of a face , denoted by , is the number of edges incident with , where each cut-edge is counted twice. A face of degree , at least or at most is called a -face, -face, or -face. A -face with consecutive vertices along its boundary in some direction is often said to be a -. Two faces are called adjacent if they are incident with a common edge.

A total -coloring of a graph is a coloring from to such that no two adjacent or incident elements have the same color. A graph is said to be totally -colorable if it admits a total -coloring. The total chromatic number of , denoted by , is the smallest integer such that is totally -colorable. The Total Coloring Conjecture (TCC) states that every simple graph is totally ()-colorable Behzad (1965); Vizing (1968), where is the maximum degree of . This conjecture has been proved for graphs with in Kostochka (1996). For planar graphs, the only open case of TCC is that of maximum degree 6; see Borodin (1989); Jensen and Toft (1995); Sanders and Zhao (1999). More precisely, if is a planar graph with , then . For planar graphs with maximum degree 7 or 8, some related results can be found in Chang et al. (2011, 2013); Du et al. (2009); Hou et al. (2011, 2008); Liu et al. (2009); Shen and Wang (2009b); Wang and Wu (2011, 2012); Xu and Wu (2014); Wang et al. (2014). Moreover, for planar graphs of maximum degree 6, it is proved that if does not contain 5-cycles Hou et al. (2011) or 4-cycles Shen and Wang (2009a). In this paper, we show that every planar graph with has a total 7-coloring if contains no some forbidden 4-cycles, which improves the result of Shen and Wang (2009a).


Fig. 1: (a) diamond (b) house
Theorem 1.1

Suppose that is a planar graph with . If does not contain a subgraph isomorphic to a diamond or a house, as shown in Figure 1, and every 6-vertex in is not incident with two adjacent 4-cycles or three cycles with sizes for some , then .

2 Reducible configurations

Let be a minimal counterexample to Theorem 1.1, in the sense that the quantity is minimum. That is, satisfies the following properties:

(1) is a planar graph of maximum degree 6.

(2) contains no subgraphs isomorphic to a diamond or a house.

(3) Every 6-vertex of is incident with neither two adjacent 4-cycles, nor three cycles with sizes for some .

(4) is not totally 7-colorable such that is minimum subject to (1),(2),(3).

Notice that every planar graph with maximum degree 5 is totally 7-colorable Kostochka (1996). Additionally, it is easy to check that every subgraph of also possesses (2) and (3). Therefore, every proper subgraph of has a total 7-coloring using the color set =. For a vertex , we use to denote the set of colors appearing on and its incident edges, and =. This section is devoted to investigating some structural information, which shows that certain configurations are reducible, i.e. they can not occur in .

Lemma 2.1

Let be an edge of such that or . Then .

The subgraph induced by all edges, whose two ends are 2-vertex and 6-vertex respectively in is a forest.

The proof of Lemma 2.1 can be found in Borodin (1989).

For any component of the forest stated in Lemma 2.1 (2), we can see that all leaves (i.e. 1-vertices) of are 6-vertices. Therefore, has a maximum matching that saturates every 2-vertex in . For each 2-vertex in , we refer to the neighbor of that is saturated by as the master of , see Borodin et al. (1997). Clearly, for a given , each 6-vertex can be the master of at most one 2-vertex, and each 2-vertex has exactly one master.

The following result follows from Lemma 2.1 directly.

Lemma 2.2

Every 4-face in is incident with at most one 2-vertex.

Lemma 2.3

Let be a 3-face incident with a 2-vertex. Then every 6-vertex incident with has only one neighbor of degree 2.

Proof: Let be the 2-vertex incident with , and be the two 6-vertices incident with . We first show that the result holds for , and then holds for analogously. Assume to the contrary that has another neighbor of degree 2, say . Let be a total 7-coloring of by the minimality of . Erase the colors on and . Without loss of generality, we assume =. If , then can be properly colored with 7. Hence, has a total 7-coloring by coloring properly (Since are 2-vertices, there are at least three available colors for each of them), and a contradiction. So we assume . Let . When , we can color with and recolor with 7. When , let and . We first exchange the colors of and , and then color with and recolor with . Therefore, we can obtain a 7-total-coloring of by coloring with two available colors. This contradicts the assumption of .

Lemma 2.4

has no (4,4,4)-face.

Proof: Suppose that has a (4,4,4)-face with three incident vertices , and . By the minimality of , has a total 7-coloring. Erase the colors on for . Clearly, each element in has at least three available colors. Since every 3-cycle is totally 3-choosable, it follows that has a total 7-coloring, and a contradiction.

Lemma 2.5

has no (3,5,3,5)-face.

Proof: Assume to the contrary that has a (3,5,3,5)-face , where , . By the minimality of , has a total 7-coloring . Erase the colors on . Clearly, each edge of has at least two available colors. Since even cycles are 2-edge-choosable, we can properly color edges and . Additionally, since , are 3-vertices and they are not adjacent (Since contains no subgraph isomorphic to a diamond), we can properly color and with two available colors. Hence, we obtain a total 7-coloring of , and a contradiction.

Lemma 2.6

Every 6-vertex incident with a 2-vertex in is adjacent to at most five -vertices.

Proof: Let be a 6-vertex incident with a 2-vertex in . Assume to the contrary that contains six -vertices. Let , where for . By the minimality of , has a total 7-coloring . Without loss of generality, we assume . Erase the colors on for . If 7 does not appear on the edges incident with , then we can properly color with 7. Otherwise, we can properly color with by recoloring with 7. Additionally, since are -vertices, there is at least one available color for each of them and by Lemma 2.1 (1) for any , . Hence, we can obtain a 7-total-coloring of , and a contradiction.

Lemma 2.7

contains no configurations depicted in Figure 2, where the vertices marked by have no other neighbors in .

Fig. 2: Three forbidden configurations in

Proof: For configuration (1), by the minimality of , has a 7-total-coloring . Without loss of generality, we assume that . If (or ), then we can properly color with 7 (or with by recoloring with 7). If , then we can properly color with by recoloring with 7. So, we assume . Let . Obviously, . If , then we can recolor with 7, with , with , and then properly color with 7. If , then . Therefore, we can safely interchange the colors of and , recolor with 7, with , with , and then properly color with 7. Thus, we obtain a 7-total-coloring of , and a contradiction.

For configuration (2), let be a 7-total-coloring of . Assume that , where for , and . By a similar argument as in (1), we assume . Let and . Obviously, . First, if , then we can recolor with 7, with , with , and then properly color with 7. Second, if , then . When , we can safely interchange the colors of and , recolor with 7, with , with , and then properly color with 7. When , we can safely interchange the colors of and , recolor with 7, with , with , and then properly color with 7. Third, if , then . When , we can recolor and with 7, recolor , and with , and then properly color with 7. When , we can safely interchange the colors of and , recolor and with 7, recolor , and with , and then properly color with 7. Hence, we obtain a 7-total-coloring of , and a contradiction.

For configuration (3), let be a 7-total-coloring of . Assume that , where for , and . By a similar argument as in (1), assume that . Let and . Obviously, and . If , then we can recolor with 7, with , with , and then properly color with 7. If , then . Therefore, we can recolor with 7, with , with , with , and then properly color with 7. So, we obtain a 7-total-coloring of , and a contradiction.

3 Discharging

In this section, to complete the proof of Theorem 1.1, we will use discharging method to derive a contradiction. For a vertex , denote by and (or simply by and ) respectively the number of 3-faces and 4-faces incident with . For a face , denote by and (or simply by and ) respectively the number of 2-vertices and 3-vertices incident with .

According to Euler’s formula , we have

Now, we define to be the initial charge of . Let for each . Obviously, . Then, we apply the following rules to reassign the initial charge that leads to a new charge . If we can show that for each , then we obtain a contradiction, and complete the proof.

(R1)

From each -vertex to each of its incident -face , transfer

, if , and is a ()-face;

, if , and is a ()-face or ()-face;

, if , and is incident with a 2-vertex or 3-vertex.

(R2)

From each 6-vertex to each of its incident 4-face , transfer

, if is a ()-face;

, if is a ()-face;

, if is a ()-face or ()-face

, if is a ()-face;

, if is a ()-face;

, if is a ()-face.

(R3)

From each 6-vertex to each of its adjacent 2-vertex , transfer

, if is incident with a 3-face;

, if is not incident with a 3-face and is a master of ;

, if is not incident with a 3-face and is not a master of .

(R4)

From each 4-face to each of its adjacent -vertex , transfer

, if and is not incident with a 3-face;

, if and is not incident with a 3-face.

(R5)

From each -face to each of its adjacent -vertex , transfer

, if and is incident with a 3-face;

, if and is not incident with a 3-face;

, if and is incident with a 3-face;

, if and is not incident with a 3-face.

(R6)

Each -face transfer to its adjacent ()-face.

(R7)

Every -face with positive charge after R1 to R6 transfers its remaining charges evenly among its incident 6-vertices.

The rest of this article is to check that for every .

4 Final charge of faces

Let be a face of . Suppose that is a 3-face. By Lemma 2.1(1) and Lemma 2.4, it follows that is incident with at most two -vertices. If is incident with at most one -vertex, then by (R1) or . If is incident with two -vertices, then by (R1) and (R6), .

Suppose that is a 4-face. Clearly, is not adjacent to a 3-face, since does not contain any subgraph isomorphic to a house. If is incident with neither a 2-vertex nor a 3-vertex, then ; If is incident with a 2-vertex, then is a (2,6,,6)-face by Lemma 2.1 (1) and Lemma 2.2, and the 2-vertex is not incident with any 3-face (Since contains no subgraph isomorphic to a diamond). So, by (R2), (R4) and (R7), (When is incident with a 3-vertex), or (When is not incident with any 3-vertex); If is not incident with a 2-vertex but is incident with a 3-vertex, then is either a (3,,,)-face, or a (3,,,)-face by Lemma 2.1 (1) and Lemma 2.5. For the former case, after (R1), (R2) and (R4), has at least (When is (3,)-face), or (When is ()-face), or (When is (3,6,4,6)-face). Therefore, by (R7). For the latter case, by (R1), (R2), (R4) and (R7) (When is (3,)-face), or by (R2), (R4) and (R7) (When is (3,)-face).

Suppose that is a 5-face. Since does not contain any subgraph isomorphic to a house, it has that every 2-vertex incident with it is not incident with a 3-face. Obviously, . If , then has at least after to (R6), and hence by . If , then is not adjacent to any ()-face. Hence, has at least after to (R6), and .

Suppose that is a 6-face. Then, at most one 2-vertex incident with is incident with a 3-face. Otherwise, contains a subgraph isomorphic to a house. By Lemma 2.1 (1) and (2), it is easy to see that and . When , the number of -faces adjacent to is at most . Therefore, has at least after to (R6), and hence by . When , it follows that is not adjacent to any -faces. Therefore, has at least after to (R6), and .

For the convenience of proving for every , we first introduce the following Lemma, which indicates that every -face has positive charges.

Lemma 4.1

Let be a 6-vertex. Then receives at least from each of its incident -face by .

Proof: Let be a ()-face incident with . Clearly, the number ()-faces adjacent to is at most .

Suppose . Then has at least charges after (R5) to (R6). Since by Lemma 2.1 (2) and by Lemma 2.1 (1). Therefore, receives at least (when ) from .

Suppose . Clearly, . Particularly, in the case of , is not adjacent to any ()-face. First, when , it has that is incident with at most two 2-vertices that are incident with a 3-face (Otherwise, there is a subgraph isomorphic to a house, and a contradiction). Therefore, has at least charges after (R5) to (R6). Second, when , it follows that . If , then is adjacent to at most three ()-faces (Note that when is adjacent to a ()-face, has to be incident with a 4-vertex. So, is incident with at most four 6-vertices in this case). Therefore, has at least charges after (R5) to (R6). If , then is not adjacent to any ()-face. Therefore, has at least charges after (R5) to (R6). Third, when , it has that . In this case, we can see that has at least charges after (R5) to (R6). All of the above show that sends at least by (R7).

By Lemma 4.1, we can see that for every -face .

4.1 Final charge of vertices

We start with an observation and a lemma.

Observation. Let be a vertex of . Since has no subgraph isomorphic to a diamond, we have . Moreover, if is a 6-vertex, then by the condition of Theorem 1.1, and .

Lemma 4.2

Suppose that is a 6-vertex incident with three consecutive faces of size 4, 6 and 4, respectively, where the 6-face is denoted by ; See Figure 3 (a). Then by , gives at least

, if is incident with at most two -vertices;

, if is incident with three -vertices and (See Figure 3 (b));

, if is incident with three -vertices and (See Figure 3 (c));

, if is incident with three -vertices and .

Fig. 3: Cases of Lemma 4.2

Proof: Since contains no subgraph isomorphic to a diamond or a house, and are not incident with a 3-face if and , and is incident with at most one 2-vertex that is incident with a 3-face.

For (1), if and , then is adjacent to at most two ()-faces. Therefore, has at least charges after (R5) to (R6), and receives at least from .

If exact one of and is a -vertex, say , then we consider two cases. First, . In this case, is adjacent to at most one ()-face. Particular, if is a 2-vertex, denoted by =, then is not adjacent to another 2-vertex that is incident with a 3-face by Lemma 2.3. This implies that when is incident with a 2-vertex that is incident with a 3-face, the 2-vertex is a neighbor of and so is not incident with ant ()-face. Consequently, has at least ( is a 2-vertex), ( is a 3-vertex)}= charges after (R5) to (R6), and receives at least from . Second, , i.e. is incident with only one -vertex . Then, is adjacent to at most three ()-faces. Therefore, has at least charges after (R5) to (R6), and receives at least from .

If and , then . When , one can readily check that is not adjacent to any ()-face. Therefore, has at least charges after (R5) to (R6), and receives at least from . When , it has that is adjacent to at most two ()-faces. Therefore, has at least charges after (R5) to (R6), and receives at least from . When , it has that has at least charges after (R5) to (R6), and receives more than from .

For (2) and (3), it follows that is not adjacent to any ()-face. If , then the other -vertex incident with is a 3-vertex. Therefore, has at least charges after (R5) to (R6), and receives at least from . If , then has at least charges after (R5) to (R6), and receives at least from .

For (4), it is clear that and are -vertices. Without loss of generality, we assume and ; See Figure 3 (d), where is another -vertex incident with . If , then is not incident with a 3-face by Lemma 2.3. Therefore, has at least charges after (R5) to (R6), and receives at least from .

In the following, we turn to the proof of for every . Let be a vertex of . By Lemma 2.1 (1), we have .

Suppose that is a 2-vertex. Then has two neighbors with degree 6 by Lemma 2.1 (1). If is incident with a 3-face, then is incident with a -face. So, receives from each of its neighbors by (R3), and receives 1 from its incident -face. Hence, . If is not incident with a 3-face, then by (R3) receives from its master and from its other neighbor of degree 6, and receives from each of its adjacent -face by (R4) and (R5). Therefore, .

Suppose that is a 3-vertex. If is incident with a 3-face, then is incident with two -faces since does not contain any subgraph isomorphic to a house. So, by . If is not incident with any 3-face, then is incident with three -faces. Hence, by (R4) and (R5), .

Suppose that is a 4-vertex. By the discharging rules (R1) to (R7), we have .

Suppose that is a 5-vertex. By the observation, we have . If , then is incident with at most five 4-faces. So, by (R1), =0. If , then by the condition of Theorem 1.1. So, by (R1), =. If =2, then by the same reason. So, by (R1) =.

Suppose that . By the observation we have +. Denote by the number of 2-vertices adjacent to . Then by Lemma 2.6. When , it is clear that by (R1) and (R2). When , denote by the unique 2-vertex adjacent to . First, . Then, by (R1), (R2) and (R3). Second, . In this case, by the condition of Theorem 1.1, either or . For the former, we have by (R1),(R2) and (R3). For the latter case, if is incident with a -face, then is incident with at most one (3,5,3,6)-face by Lemma 2.6. Therefore, by (R2) and (R3); If is not incident with a -face, then is incident with at most two (3,5,3,6)-faces. Therefore, by (R2) and (R3). In what follows, we assume , and then by Lemma 2.3, we have that every 2-vertex is not incident with a 3-face. Thus, when +, by (R3). Now, we further consider the following three cases.

Case 1. +. If and , then by Lemma 2.3. Therefore, by (R1) and (R3). If and , denoted by the 4-face incident with , then by Lemma 2.6 and (R2),(R3), when is a (2,6,3,6)-face, it has that and ; When is a ()-face, and ; When is a ()-face, and