Size of edge-critical uniquely 3-colorable planar graphs 1footnote 11footnote 1Supported by 973 Program of China 2013CB329601, 2013CB329603, National Natural Science Foundation of China Grant 61309015 and National Natural Science Foundation of China Special Equipment Grant 61127005.

Size of edge-critical uniquely 3-colorable planar graphs 111Supported by 973 Program of China 2013CB329601, 2013CB329603, National Natural Science Foundation of China Grant 61309015 and National Natural Science Foundation of China Special Equipment Grant 61127005.

Zepeng Li lizepeng@pku.edu.cn Enqiang Zhu zhuenqiang@pku.edu.cn Zehui Shao zshao@cdu.edu.cn Jin Xu jxu@pku.edu.cn Key Laboratory of High Confidence Software Technologies, Peking University, Beijing, 100871, China Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, China School of Information Science and Technology, Chengdu University, Chengdu, 610106, China
Abstract

A graph is uniquely k-colorable if the chromatic number of is and has only one -coloring up to permutation of the colors. A uniquely -colorable graph is edge-critical if is not a uniquely -colorable graph for any edge . Mel’nikov and Steinberg [L. S. Mel’nikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] asked to find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if is such a graph with vertices, then , which improves the upper bound given by Matsumoto [N. Matsumoto, The size of edge-critical uniquely 3-colorable planar graphs, Electron. J. Combin. 20 (3) (2013) P49]. Furthermore, we find some edge-critical 3-colorable planar graphs which have vertices and edges.

keywords:
planar graph; unique coloring; uniquely -colorable planar graph; edge-critical
Msc:
05C15
journal:

1 Introduction

A graph is uniquely k-colorable if and has only one -coloring up to permutation of the colors, where the coloring is called a unique -coloring. In other words, all -colorings of induce the same partition of into independent sets. In addition, uniquely colorable graphs may be defined in terms of their chromatic polynomials, which initiated by Birkhoff Birkhoff1912 () for planar graphs in 1912 and, for general graphs, by Whitney Whitney1932 () in 1932. Because a graph is uniquely -colorable if and only if its chromatic polynomial is . For a discussion of chromatic polynomials, see Read Read1968 ().

Let be a uniquely -colorable graph, is edge-critical if is not uniquely -colorable for any edge . Uniquely colorable graphs were defined and studied firstly by Harary and Cartwright Harary1968 () in 1968. They proved the following theorem.

Theorem 1.1.

(Harary and Cartwright Harary1968 ()) Let be a uniquely -colorable graph. Then for any unique -coloring of , the subgraph induced by the union of any two color classes is connected.

As a corollary of Theorem 1.1, it can be seen that a uniquely -colorable graph has at least edges. Furthermore, if a uniquely -colorable graph has exactly edges, then is edge-critical. There are many references on uniquely colorable graphs. For example see Chartrand and Geller Chartrand1969 (), Harary, Hedetniemi and Robinson Harary1969 () and Bollobás Bollob¨¢s1978 ().

Chartrand and Geller Chartrand1969 () in 1969 started to study uniquely colorable planar graphs. They proved that uniquely 3-colorable planar graphs with at least 4 vertices contain at least two triangles, uniquely 4-colorable planar graphs are maximal planar graphs, and uniquely 5-colorable planar graphs do not exist. Aksionov Aksionov1977 () in 1977 improved the low bound for the number of triangles in a uniquely 3-colorable planar graph. He proved that a uniquely 3-colorable planar graph with at least 5 vertices contains at least 3 triangles and gave a complete description of uniquely 3-colorable planar graphs containing exactly 3 triangles.

For an edge-critical uniquely -colorable planar graph , if , then it is easy to deduce that is tree and has exactly edges. If , then is a maximal planar graph and has exactly edges by Euler’s Formula. Therefore, it is sufficient to consider the size of uniquely -colorable planar graphs. We denote by the set of all edge-critical uniquely -colorable planar graphs and by the upper bound of the size of edge-critical uniquely -colorable planar graphs with vertices.

In 1977 Aksionov Aksionov1977 () conjectured that . However, in the same year, Mel’nikov and Steinberg Mel'nikov1977 () disproved the conjecture by constructing a counterexample , which has 16 vertices and 30 edges. Moreover, they proposed the following problems:

Problem 1.2.

(Mel’nikov and Steinberg Mel'nikov1977 ()) Find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with vertices. Is it true that for any ?

Recently, Matsumoto Matsumoto2013 () constructed an infinite family of edge-critical uniquely 3-colorable planar graphs with vertices and edges, where . He also gave a non-trivial upper bound for .

In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs with vertices and improve the upper bound of given by Matsumoto Matsumoto2013 () to , where . Moreover, we give some edge-critical 3-colorable planar graphs which have vertices and edges. It follows that the conjecture of Mel’nikov and Steinberg Mel'nikov1977 () is false because if .

2 Notation

Only finite, undirected and simple graphs are considered in this paper. For a planar graph , , and are the sets of vertices, edges and faces of , respectively. We denote by and the minimum degree and maximum degree of graph . The degree of a vertex , denoted by , is the number of neighbors of in . The degree of a face , denoted by , is the number of edges in its boundary, cut edges being counted twice. When no confusion can arise, and are simplified by and , respectively. A face is a -face if and a k-face if . The similar notation is used for cycles. We denote by the set of vertices of with degree and by the set of vertices of with degree at least , where . The similar notation is used for the set of faces of .

A -wheel is the graph consists of a single vertex and a cycle with vertices together with edges from to each vertex of . A planar (resp. outerplanar) graph is maximal if is not planar (resp. outerplanar) for any two nonadjacent vertices and of . Let and be two disjoint subset of , we use to denote the number of edges of with one end in and the other in . In particular, if or , we simply write or for , respectively. To contract an edge of a graph is to delete the edge and then identify its ends. The resulting graph is denoted by . Two faces and of are adjacent if they have at least one common edge. A -cycle is said to be a separating -cycle in if the removal of disconnects the graph .

A k-coloring of is an assignment of colors to such that no two adjacent vertices are assigned the same color. Naturally, a -coloring can be viewed as a partition of , where denotes the set of vertices assigned color , and is called a color class of the coloring for any . Two -colorings and of are said to be distinct if they produce two distinct partitions of into color classes. A graph is k-colorable if there exists a -coloring of , and the chromatic number of , denoted by , is the minimum number such that is -colorable.

The notations and terminologies not mentioned here can be found in Bondy2008 ().

3 Properties of edge-critical uniquely -colorable planar graphs

Let be a 3-colorable planar graph and be a 3-coloring of . It is easy to see that the restriction of to is a 3-coloring of , where . For convenience, we also say is a 3-coloring of . If there exists a 3-coloring of such that , then we say that can be extended to a 3-coloring of .

Theorem 3.1.

Let be a uniquely 3-colorable planar graph. Then if and only if is 3-colorable for any edge .

Proof Suppose that , then, by definition, has at least two distinct 3-colorings for each . Since is uniquely 3-colorable, we conclude that there exists a 3-coloring of such that . Hence is 3-colorable.

Conversely, suppose that . Then there exists an edge such that is also a uniquely 3-colorable planar graph. Obviously, for any unique 3-coloring of , we have . So is not 3-colorable. This establishes Theorem 3.1. ∎

The following result is obtained by Theorem 3.1.

Corollary 3.2.

Let and . If is incident with exactly one 4-face and all other faces incident with are triangular, then is even.

Proof Suppose that the result is not true. Let be the neighbors of and and be the vertices of the 4-face. Then the graph contains a -wheel. Hence is not 3-colorable, a contradiction with Theorem 3.1. ∎

Theorem 3.3.

Suppose that and is a subgraph of . If is uniquely 3-colorable, then we have

(i)

;

(ii)

For any vertex , .

Proof (i) Suppose that , then there exists an edge such that is also uniquely 3-colorable. Let be a unique 3-coloring of . Since , then has a 3-coloring which is distinct from . Note that , we have . Thus, can be extended to a 3-coloring of . So has two distinct 3-colorings and , which contradicts .

(ii) Suppose that there exists a vertex such that . Let be a unique 3-coloring of and be the three neighbors of in . Then their exist at least two vertices among and receive the same color. We assume w.l.o.g. that . Since , then has a 3-coloring which is distinct from . Note that is uniquely 3-colorable and , we have . Thus, can be extended to a 3-coloring of . This is a contradiction. ∎

Corollary 3.4.

Suppose that contains a sequence of triangles satisfying and have a common edge, where and . Let and be the vertices in and , respectively, then and .

Proof Let be the neighbors of in . Since the subgraph of consists of triangles is uniquely 3-colorable, by Theorem 3.3, we know that is not adjacent to or in . Thus, . Similarly, since the subgraph of consisting of triangles is uniquely 3-colorable, we have . ∎

By Corollary 3.4, we have the following result.

Corollary 3.5.

Suppose that has no separating 3-cycles. Let be a subgraph of that consists of a sequence of triangles such that each has a common edge with for some , where . Then is a maximal outerplanar graph.

For a planar graph , if has no separating 3-cycles, we call the subgraph in Corollary 3.5 a triangle-subgraph of . Note that a triangle is a triangle-subgraph of . Therefore, any has at least one triangle-subgraph. A triangle-subgraph of is maximal if there is no maximal outerplanar subgraph of such that . In other words, the graph consists of the longest sequence of triangles such that each has a common edge with for some .

Theorem 3.6.

Suppose that has no separating 3-cycles. Let be a uniquely 3-colorable subgraph and be any two maximal triangle-subgraphs of . If , , then we have

(i)

and have at most one common vertex;

(ii)

If and have a common vertex , then ; otherwise, ;

(iii)

If and have a common vertex , and have a common vertex and , , then the union of and is uniquely 3-colorable.

Proof Let be a unique 3-coloring of .

(i) Suppose, to the contrary, that and have two common vertices and . Since , then and are not adjacent in both and . Otherwise, if , this contradicts Corollary 3.4; if , then is uniquely 3-colorable but not edge-critical, a contradiction with Theorem 3.3. By the definition of a triangle-subgraph, we know that there exists a sequence of triangles in such that and have a common edge and , , where .

If . Let be a neighbor of in , then . Since , then has a 3-coloring which is distinct from . Note that both and the subgraph of consists of triangles are uniquely 3-colorable and . So , , namely . Therefore, can be extended to a 3-coloring of which is distinct from . This contradicts .

If . Let be a neighbor of in satisfying . Since , then has a 3-coloring which is distinct from . Since both and the subgraph of consists of triangles are uniquely 3-colorable, we have . Therefore, can be extended to a 3-coloring of which is distinct from . It is a contradiction.

(ii)Case 1. and have a common vertex .

Suppose that and are two edges with and . If there exists a vertex such that , we assume w.l.o.g. that , then . Since , has a 3-coloring which is distinct from . Note that both and are uniquely 3-colorable, we have and . Thus and then can be extended to a 3-coloring of which is distinct from . If for any , then . Thus, we have either and , or and . Since has a 3-coloring which is distinct from , and and are uniquely 3-colorable, we have . Therefore, can be extended to a 3-coloring of which is distinct from .

Case 2. and have no common vertex.

Suppose that and are 4 edges with and , . Then there exist two edges, say and , such that . By using a similar argument to Case 1, we can obtain a 3-coloring of , which is distinct from and can be extended to a 3-coloring of . It is a contradiction.

(iii) By definition of , there exists a sequence of triangles in such that and have a common edge and , , where .

Suppose that , . Let be an arbitrary neighbor of in . Then . Since , has a 3-coloring which is distinct from . Note that and the subgraph of consists of triangles are uniquely 3-colorable, we have and . Therefore, can be extended to a 3-coloring of which is distinct from . This contradicts .

Suppose that , then their exists such that . We assume w.l.o.g. that . Let be a neighbor of in satisfying . If , then . Since , has a 3-coloring which is distinct from . Note that and the subgraph of consists of triangles are uniquely 3-colorable, we have , and . Thus, . Therefore, can be extended to a 3-coloring of which is distinct from . This contradicts . If, then . Since , has a 3-coloring which is distinct from . Since and the subgraph of consists of triangles are uniquely 3-colorable, we have and . Thus, . This contradicts .

Suppose that . Using the fact that any coloring of two vertices with can be extended uniquely to a 3-coloring of , we can obtain that the union of and is uniquely 3-colorable, where . ∎

4 Size of edge-critical uniquely -colorable planar graphs

In this section, we consider the upper bound of for edge-critical uniquely 3-colorable planar graphs with vertices.

Suppose that and has no separating 3-cycles. Let be all of the maximal triangle-subgraphs of . For two maximal triangle-subgraphs and having a common vertex , if there exists such that and satisfy the condition of Case (iii) in Theorem 3.6, namely and have a common vertex (say ), and have a common vertex (say ) and , then we say that and satisfy Property P. Let . (We will use such notation without mention in what follows.) Now we analyse the relationship between , the number of 4-faces of , and . For a vertex , we use to denote the number of maximal triangle-subgraphs of that contain .

First we construct a new graph from with , , , where in corresponds to in for any . The edges in are constructed by the following two steps.
Step 1: For every with , add the edge to if both and contain . (see e.g. Fig. 1)
Step 2: For every with , let contain and they appear in clockwise order around . For any with and satisfying Property P, then add the edge to .

Let be the subgraph of with vertex set and edge set .

Then we add some edges in to such that the resulting graph, denoted by , is connected and has the minimum number of edges. Now the construction of the edges of the graph is completed. (see e.g. in Fig. 1, we first join the edges , and , then join the edges , , , and .)

Figure 1: An example of a graph and the corresponding graph .

Remark. By the definition of , if , then and must have a common vertex. For a edge-critical uniquely 3-colorable graph , if for any , then the graph obtained by above construction is unique; otherwise, is not unique. Furthermore, we have Theorem 4.1.

Theorem 4.1.

Suppose that has no separating 3-cycles. Let , , be all of the maximal triangle-subgraphs of , then is a simple planar graph and .

Proof By Corollary 3.5 and Theorem 3.6(i), we know that has no loops or parallel edges. So is a simple graph. Note that is a planar graph. For any with and any with and satisfying Property P, then is a planar graph and there exist no edges such that and , where and the subscripts are taken modulo . Now we prove that is a forest. If , then, by the definition of and Theorem 3.6(iii), we know that there exist and such that the graph is uniquely 3-colorable. Thus, and does not satisfy Property P, namely . Therefore, is a forest. By the definition of , it is easy to see that is a tree. By the definition of , we can conclude that is a planar graph.

For any distinct faces and of , by the definition of , it can be seen that there exist two distinct 4-faces of corresponding to and , respectively. Conversely, for any 4-face of , let be all of the maximal triangle-subgraphs satisfying and have common edges, . Let be the common vertex of , because is tree, there exists a unique face of incident with . Thus, . ∎

Theorem 4.2.

Suppose that has no separating 3-cycles. Let , , be a sequence of faces in such that and are adjacent, , . If and , , let be all of the vertices incident with the faces . Then and is uniquely 3-colorable, where denotes the set of the vertices incident with .

Proof The proof is by induction on . Let and be the vertices incident with , but not incident with , , . If , since , by Theorem 3.6 (iii), we know that is uniquely 3-colorable. Suppose that . By hypothesis, is uniquely 3-colorable. Since , by Theorem 3.6 (i), we have , namely . Therefore, by Theorem 3.6 (iii), we obtain that is uniquely 3-colorable. ∎

For a planar graph , let and be two cycles of . and are dependent if there exists a sequence of cycles of such that and have common edges and , where , . Obviously, if and have a common edge, then they are dependent.

Lemma 4.3.

Let be a planar graph, . If any -cycle of is dependent with at most 3-cycles for and with at most 3-cycles for , then .

Proof The proof is by contradiction. Let be a smallest counterexample to the lemma, then satisfies the conditions of the lemma and . Suppose that is not connected, let be a connected component of . If and , it is easy to see that . This is a contradiction. Otherwise, we assume w.l.o.g. that . Since any -cycle of is dependent with at most 3-cycles for and with at most 3-cycles for , the same is true of and . By the minimality of , we have . Furthermore, if , then ; otherwise, . Therefore, , a contradiction.

Suppose that is connected. If contains a cut vertex , let be the vertex sets of the connected components of , respectively, and , . Obviously, satisfies the conditions of the lemma. If , then, by the minimality of , ; otherwise, , . Therefore, , a contradiction.

Now we assume that is 2-connected. If contains no 3-faces, then . Thus . By Euler’s Formula, we have . This contradicts the choice of . If contains exactly one 3-face, then contains at least one 5-face. Thus, and then . If contains at least two 3-faces, then each 3-face is dependent with at least two 5-faces for is 2-connected. We claim that , namely .

For any face , we set the initial charge of to be . We now use the discharging procedure, leading to the final charge , defined by applying the following rule:

RULE. Each 3-face receives from each dependent 5-face.

For any face , if , since is dependent with at least two 5-faces, then . If , then . If , then . If , then, by hypothesis, . Therefore,