Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable

Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable

Seog-Jin Kim and Xiaowei Yu Department of Mathematics Education, Konkuk University, Korea. Email: skim12@konkuk.ac.kr
Supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2015R1D1A1A01057008).School of Mathematics, Shandong University, Jinan, Shandong, 250100, P. R. China. Corresponding author. Email: xwyu2013@163.com
Supported by the National Natural Science Foundation of China (11371355, 11471193, 11271006, 11631014), the Foundation for Distinguished Young Scholars of Shandong Province (JQ201501), and fundamental research funding of Shandong University.
Abstract

DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which is an extension of the two results above.

Keywords: Coloring, list-coloring, DP-coloring, signed graph

1 Introduction

We use standard notation. For a set , Pow denote the power set of , i.e., the set of all subsets of . We denote by the set of integers from to . All graphs considered here are finite, undirected, and simple. For a graph , , , and denote the vertex sets, edge sets and face sets of , respectively. For a set , is the subgraph of induced by .

Recall that a proper -coloring of a graph is a mapping such that for any . The minimum integer such that admits a proper coloring is called the chromatic number of , and denoted by .

List coloring is a generalization of graph coloring that was introduced independently by Vizing [15] and Erdős, Rubin, and Taylor [8]. Let be a set of colors. A list assignment of is a mapping that assigns a set of colors to each vertex. If for all , then is called a -list assignment. A proper coloring is called an -coloring of if for any . The list-chromatic number or the choice number of , denoted by , is the smallest such that admits an -coloring for every -list assignment for .

Since a proper -coloring corresponds to an -coloring with for any , we have . It is well-known that there are infinitely many graphs satisfying , and the gap can be arbitrarily large.

In order to consider some problems on list chromatic number, Dvořák and Postle [7] considered a generalization of a list coloring. They call it a correspondence coloring, but we call it a DP-coloring for short, following Bernshteyn, Kostochka and Pron [4].

Let be a graph and be a list assignment of . For each edge in , let be an arbitrary matching (maybe empty) between and . Without abuse of notation, we sometimes regard as a bipartite graph in which the edges are between and , and the maximum degree is at most 1.

Definition 1.1

Let , which is called a matching assignment over . Then a graph is said to be the -cover of if it satisfies all the following conditions:

1. The vertex set of is .

2. For every , the graph is a clique.

3. For any edge in , and induce the graph obtained from in .

Definition 1.2

An -coloring of is an independent set in the -cover with . The DP-chromatic number, denoted by , is the minimum integer such that admits an -coloring for each -list assignment and each matching assignment over . We say that a graph is DP--colorable if .

Note that when is a simple graph and

 ML,uv={(u,c)(v,c):c∈L(u)∩L(v)}

for any edge in , then admits an -coloring if and only if admits an -coloring. This implies . Thus DP-coloring is a generalization of the list coloring. Thus, given the fact , it is interesting to check whether or not.

Dvořák and Postle [7] showed that if is a planar graph, and if is a planar graph with girth at least 5. Also, Dvořák and Postle [7] observed that if is -degenerate.

On the other hand, there are some differences between DP-coloring and list coloring. There are infinitely many simple graphs satisfying : It is known that for each even integer (see [4]). Furthermore, the gap can be arbitrary large. For example, Bernshteyn [2] showed that for a simple graph with average degree , we have , while Alon [1] proved that and the bound is sharp. Recently, there are some other works on DP-colorings, see [3, 5, 10].

Thomassen [14] showed that every planar graph is 5-choosable, and Voigt [16] showed that there are planar graphs which are not 4-choosable. Thus finding sufficient conditions for planar graphs to be 4-choosable is an interesting problem.

Two faces are adjacent if they have at least one common edge, and two faces are normally adjacent if they are adjacent and have exactly one common edge. Let be the cycle of length . Lam, Xu, and Liu [12] verified that every planar graph without is 4-choosable. And Cheng, and Chen, Wang [6], and Kim and Ozeki [11] extended the result independently by certifying the following theorem.

Theorem 1.3

The following results hold independently.

A.

([6]) If is a planar graph without -cycles adjacent to -cycles, then .

B.

([11]) If is a planar graph without -cycles, then .

In this paper, we extend Theorem 1.3 by proving the following theorem.

Theorem 1.4

If is a planar graph without -cycles adjacent to -cycles, then .

Even though for some special graphs, Theorem 1.4 is not trivial from Theorem 1.3 (A). We will give an explanation in section 3.1.

2 Proof of Theorem 1.4

Suppose that Theorem 1.4 does not hold. In the rest of paper, let be a minimal counterexample to Theorem 1.4 with fewest edges. From our hypothesis, graph has the following properties:

1. is connected; and

2. has no subgraph isomorphic to 4-cycles adjacent to 3-cycles; and

3. is not DP--colorable; and

4. any proper subgraph of is DP--colorable.

Embedding into a plane, we obtain a plane graph where are the sets of vertices, edges, and faces of , respectively.

For a vertex , the degree of in is denoted by . A vertex of degree (at least , at most , respectively) is called a -vertex (-vertex, -vertex, respectively). The notions of -face, -face, -face are similarly defined. According to (b), if a -face is adjacent to a -face, then they are normally adjacent.

For a face , if the vertices on in a cyclic order are , then we write , and call a -face.

2.1 Structures

Using the properties of above, we can obtain several local structures of .

Lemma 2.1

Graph has no -vertex.

Proof. Suppose to the contrary that there exists a -vertex in . Let be a list assignment of with for any , and let be a matching assignment over . Let and for . According to (d), admits an -coloring. Thus there is an independent set in -cover with . For , we define that

 L∗(w)=L(w)∖⋃uw∈E(G){c′∈L(w):(u,c)(w,c′)∈ML,uw and (u,c)∈I′}.

Since and is a -vertex, we have that . We denote by the restriction of into and . Obviously, is an independent set in the -cover with , which is a contradiction to (c). ∎

Next we define several different -faces. Let be a -face of .

1. If is a -face, then we call a small 5-face. Assume that is adjacent to a -face with the common edge , then we call a source of . Equivalently, the face is called a sink of (see in Figure 1).

2. If is a -face, and is incident to four -faces and one -face, and the -face is incident to the -vertex on , then we call a bad -face (see in Figure 1).

3. If is a -face, and is incident to five -faces, then we call a special -face. Meanwhile, we call the -vertex on a special vertex (see in Figure 1).

Remark 2.2
1. A special vertex is incident to at least one special -face, and a special -face is incident to exactly one special vertex.

2. If there exist two -vertices on a -face , then is neither special nor bad.

Lemma 2.3

Every source is a -vertex.

Proof. Let be a small 5-face, and let be a source of . By Lemma 2.1, there exists no -vertex in , thus we suppose that . Without loss of generality, we may assume that is adjacent to and . Let be the subgraph of induced by . Let be a list assignment of with for all , and let be a matching assignment over . Consider subgraph and for . By (d), admits an -coloring. Thus there is an independent set in the -cover with . For , we define

 L∗(v)=L(v)∖⋃uv∈E(G){c′∈L(v):(u,c)(v,c′)∈ML,uv and (u,c)∈I′}.

Because for all , we have that , and for . We denote by the restriction of into and .

Claim 2.4

The -cover has an independent set with .

Proof. Because for , it holds that and , we can color by such that has at least two available colors. By coloring greedily in order, we can find an independent set with . This completes the proof of Claim 2.4. ∎

Thus admits an -coloring such that . It implies that is DP-4-colorable. This is contradiction to (c). Thus .∎

Lemma 2.5

Let be a -vertex of . Then the following hold:

(1) is incident to at most -faces; and

(2) if is incident to -faces and , then is incident to at most special -faces.

Proof. Note that (1) holds obviously from (b).

And from the definition of a special -face, if is a -vertex of , then each of the special -faces incident to is adjacent to two -faces, and both of the two -faces are incident to , which implies (2). ∎

Lemma 2.6

Let be a -vertex of . Assume that is incident to a special or bad -face and a -face such that and are adjacent, then has no sink adjacent to .

Proof. Let be a special or a bad -face. According to the definition of special -face and bad -face, there are at least four 3-faces adjacent to . Assume that , and is adjacent to (see Figure 2). Suppose to the contrary that has a sink adjacent to with a common edge , then is a source of . However, is a -vertex, contrary to Lemma 2.3. ∎

Lemma 2.7

Let and be two bad -faces. Then they can not normally adjacent with one common edge , where is the -vertex on and .

Proof. Assume that two bad -faces, say and , are normally adjacent, and they have one common -vertex and one common edge (see Figure 3). Since both and are bad, then is a -vertex, and and are two -faces. Hence and are adjacent, contrary to (b). Thus the assumption is false. ∎

2.2 Discharging

It follows that

 ∑v∈V(2dG(v)−6)+∑f∈F(dG(f)−6)=−12

from Euler’s formula and the equality . Now we define an initial charge function for each by letting for each and for each . We are going to design several discharging rules. Since the sum of total charge is fixed during the discharging procedure, if we can change the initial charge function to the final charge function such that for each , then

 0≤∑x∈V∪Fch(x)=∑x∈V∪Fch′(x)=−12,

which is a contradiction. It means that no counterexample to Theorem 1.4 exists. Thus Theorem 1.4 holds.

Before designing the discharging rules, we define a configuration as shown in Figure 4. In Figure 4, the degrees of white vertices are at least the number of edges incident to them, but the degrees of the black vertices are equal to the number of edges incident to them. The red vertex is a special -vertex. In this configuration, is a special -face, is a bad -face, is a -face satisfying the following:

• is neither special nor bad; and

• is incident to a special -vertex , and is incident to a bad -face ; and

• is normally adjacent to .

We call the family of , that is, is a family of -faces that have the same properties of .

Remark 2.8

Let be a vertex on as shown in Figure 4. If , then is incident to at most one -face according to (b).

Discharging Rules:

R1

Each -vertex gives to each incident -face.

R2

Each -vertex gives to each incident -face.

R3

Let be a -vertex incident to at most one -face. If is incident to exactly one -face and one -face, then gives to each incident -face; Otherwise, gives to each incident -face.

R4

By Remark 2.2 (1), a special -face is exactly incident to one special vertex.

R4.1

Each special -vertex gives to each incident special -face.

R4.2

Each special -vertex gives to each incident bad -face (see in Figure 4).

R4.3

If a special -vertex is incident to three -faces, which are special -face , bad -face , and non-special or non-bad -face respectively, then gives to . (See Figure 4)

R5

Each non-special -vertex and each -vertex gives to each incident bad -face.

R6

Each source gives to each of its sinks.

R7

If is a -vertex, or is a non-special 5-vertex, or is a special -vertex but is not incident to a bad -face, then gives to each incident non-special or non-bad -face.

To complete the proof of Theorem 1.4, it remains to check that the final charge of every element in is nonnegative. This will be shown by the following two claims.

Claim 2.9

It holds that for all .

Proof. According to Lemma 2.1, there exists no -vertex in , thus we need to consider the -vertices in the following.

Case 1: When .

In this case, we have that . According to Lemma 2.3, has no any sink. If is incident to two -faces, then by R1.

And, if is incident to exactly one -face, then is incident to at most one -face according to (b). Assume that is incident to one -face, then it is incident to at most two -faces. Thus by R1, R2 and R3. Otherwise, is incident to at most three -faces. Therefore, by R1 and R3.

If is not incident to any -face, then by R2 and R3.

Case 2: When .

In this case, we have that . Assume that is special, that is, is incident to a special -face by Remark 2.2 (1). From Lemma 2.5 and the definition of special -face, it holds that is incident to exactly two -faces and one special -face. According to Lemma 2.6, has no sink. If is incident to a bad -face, then is incident to at most one bad -face by Lemma 2.7. We have that from R1 and R4. Otherwise is not incident to any bad -face. It holds that by R1, R4 and R7.

We next assume that is not special, that is, is not incident to a special -face by Remark 2.2 (1). The vertex is incident to at most two -faces by Lemma 2.5 (1). According to Lemma 2.7, is incident to at most two bad -faces.

• If is incident to two -faces, then is incident to at most one bad -face from Lemma 2.7, the definition of bad -face and (b), and is not incident to any -face by (b). Thus we have that from R1, R4 and R7.

• If is incident to exactly one -face, then is incident to at most two bad -faces by the definition of bad -face. Thus the final charge from R1, R4, R6 and R7.

• If is not incident to any -face, then is not incident to bad -face. It holds that by R2 and R7.

Case 3: When .

Note that we have that . According to Lemma 2.5, is incident to at most three -faces and at most three special -faces.

• Assume that is incident to three -faces. If is incident to three special -faces, then has no any sink by Lemma 2.6. Thus we have that from R1, R4.1. Otherwise is incident to at most two special -faces. Then has no any sink by Lemma 2.6. Hence we have that from R1, R4.1 and R5.

• Assume that is incident to at most two -faces, then is incident to at most one special -faces by Lemma 2.5 (2). If is incident to one special -faces, then from R1, R4.1, R5 and R6. Otherwise, by Lemma 2.7, is incident to at most two bad -faces. Therefore, it holds that from R1, R4.1, R5, R6 and R7.

Case 4: When .

In this case, we have that . By Lemma 2.5, is incident to at most three -faces and at most two special -faces. The smallest final charge from R1, R4.1, R5 and R6.

Case 5: When .

Observe that . By Lemma 2.5, is incident to at most -faces and at most special -faces. It holds that by R1, R4.1 and R6. ∎

Claim 2.10

It holds that for all .

Proof. Let be a face of . Because is simple, has no loops and multi-edges. Thus . If , no charge is discharged from or to , thus . If , then every vertex incident to gives to according to R1. Therefore, we have that . If , then every vertex incident to gives to according to R2. Hence the final charge . Next we assume that . Note that .

Case 1: Assume that is small, that is, all the vertices incident to are -vertices.

For , let be the number of -vertices which are incident to two 3-faces. Then has -vertices which are incident to at most one 3-face, and has at least sources. Thus we have for every by R3 and R6.

Case 2: When is a -face.

Denote the -vertex by . If is special, then according to R4.1. Next let be a non-special -face.

• Assume that is a special -vertex and is incident to a bad -face (see Figure 4). If is bad, then according to R4.2 and R3. Otherwise is not bad, then . Hence is incident to at least two -vertices which are incident to at most one -face, and are not incident to a 3-face and a 4-face at the same time by Remark 2.8, respectively. Therefore, we have that according to R3, R4.3 and R7.

• Otherwise, is a -vertex, or is not a special -vertex, or is a special -vertex and is not incident to a bad -face. If is bad, then there exists a -vertex incident to , and the -vertex is incident to at most one -face. Thus we have that according to R3 and R5. Otherwise is not bad. Hence there are at least two -vertices incident to , and each of them is incident to at most one -face. We can conclude that according to R3 and R7.

Case 3: When there exist at least two -vertices on . From Remark 2.2 (2), we have that is neither special nor bad.

• If , then by R3, R4.3 and R7.

• Otherwise , that is, is not incident to a special -vertex which is on a special -face and is incident to a bad -face. We can conclude that according to R7. ∎

The proof of Theorem 1.4 is completed. ∎

3 Remarks

3.1 Difference between DP-coloring and list coloring

A -graph is a graph consisting of two 3-vertices and three pairwise internally disjoint paths between the two 3-vertices. A -subgraph of is an induced subgraph that is isomorphic to a -graph. We use to denote such a special -subgraph of in which one of the ends of the internal chord is a -vertex and all of the other vertices are 4-vertices in .

Let be a planar graph without 4-cycles adjacent to 3-cycles. In the proof of Theorem 1.3 (A), the authors showed that if is not -choosable with fewest vertices, then contains no subgraph isomorphic to (see Lemma 4 in [6]). But we can not claim that if is not DP--colorable with fewest vertices, then contains no subgraph isomorphic to . Next we give an explanation.

In Figure 5 (a), assume that , for and . Let be the subgraph of induced by . Let be a list assignment of with for all , and let be a matching assignment over . Set and for . By (d), admits an -coloring. Thus there is an independent set in the -cover with . For , we define

 L∗(v)=L(v)∖⋃uv∈E(G){c′∈L(v):(u,c)(v,c′)∈ML,uv and (u,c)∈I′}.

Because for all , we have that , and for . We denote by the restriction of into and .

Next we give a -cover as shown in (b) of Figure 5. But we cannot find an independent set with in -cover. Thus we cannot claim that if is not DP--colorable with fewest vertices, then contains no subgraph isomorphic to . Therefore, Theorem 1.4 is not trivial from Theorem 1.3 (A).

3.2 Relationship with signed coloring

There is a concept of signed coloring of signed graphs, which was first defined by Zaslavsky [17] with slightly different form, and then modified by Máčajová, Raspaud, and Škoviera [13] so that it would be a natural extension of an ordinary vertex coloring. For detail story about signed coloring, we refer readers to [9, 11, 13].

An interesting obervation in [11] is that the signed coloring of a signed graph is a special case of a DP-coloring of . Thus Theorem 1.4 implies the following corollary, which is an extension of the result in [9].

Corollary 3.1

A graph is signed -choosable if is a planar graph without -cycle adjacent to a -cycle.

References

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