Journal of Applied Mathematics

Journal of Applied Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 849748 | 7 pages | https://doi.org/10.1155/2014/849748

General Vertex-Distinguishing Total Coloring of Graphs

Academic Editor: Tak-Wah Lam
Received24 Apr 2014
Accepted14 Jul 2014
Published03 Aug 2014

Abstract

The general vertex-distinguishing total chromatic number of a graph is the minimum integer , for which the vertices and edges of are colored using colors such that any two vertices have distinct sets of colors of them and their incident edges. In this paper, we figure out the exact value of this chromatic number of some special graphs and propose a conjecture on the upper bound of this chromatic number.

1. Introduction

All graphs considered in this paper are simple and finite. For a graph , we denote by , , , and the sets of vertices, edges, maximum degree, and minimum degree of , respectively. For a vertex of , is the degree of in . For any , we use to denote the subgraph induced by . For any undefined terms, the reader is referred to the book [1].

The coloring problem of graphs is one of the classical research areas in graph theory. It has been widely applied to various fields, such as large scheduling [2], assignment of radio frequency [3], and separating combustible chemical combinations [4]. Due to its extensive application, many new variants of colorings have been studied [5].

Recall that a -edge coloring of a graph is a mapping : , where is a set of colors. An edge coloring is proper if adjacent edges receive distinct colors. In 1985, Harary and Plantholt [6] first considered point-distinguishing chromatic index, which is a variant of edge coloring. After that, many other variants of edge coloring were introduced, such as vertex-distinguishing proper edge coloring [7], adjacent vertex-distinguishing edge coloring [8], and general adjacent vertex-distinguishing edge coloring [9].

A total -coloring of a graph is a coloring of using colors. A total -coloring is proper if no two adjacent or incident elements receive the same color. The minimum number of colors required for a proper total coloring of is called the total chromatic number of and is denoted by . Behzad [10] and Vizing [11] independently made the conjecture that, for any graph , This is known as the total coloring conjecture () and is still unproven.

Let be a total -coloring of . The total color set (with respect to ) of a vertex is the set, denoted by , of colors of and its incident edges. We denote by the set of total color sets of all vertices of . Furthermore, let be a subset of ; we use to denote the set of colors of elements of .

Like edge coloring, total coloring also has some variants. In 2005, Zhang et al. [12] added a restriction to the definition of total coloring and proposed a new type of coloring defined as follows.

Definition 1. Let be a proper total -coloring of a graph . If, for all , , then is called an adjacent vertex-distinguishing total -coloring of , or a of for short. The minimum number for which has a is the adjacent vertex-distinguishing total chromatic number of , denoted by .

Zhang et al. [12] conjectured that, for any graph , it has In [1315], authors proved that there exists a 6- of graphs with , which indicates conjecture (2) holds for such graphs. For further research on adjacent vertex-distinguishing total chromatic number, one may refer to [1623].

For a - of a graph , if is required for any two distinct vertices , then is called a vertex-distinguishing total -coloring of , abbreviated as . The minimum number such that has a is called the vertex-distinguishing total chromatic number, denoted by [24]. Zhang et al. conjectured in [24] that, for any graph , it follows that where .

In this paper, we introduce a variant of vertex-distinguishing total coloring of a graph , which relaxes the restriction that the coloring is proper. We now present the detailed definition as follows.

Definition 2. Let be a graph and be a positive integer. A total coloring of using colors is called a general vertex-distinguishing total -coloring of or of briefly if, for all , . The minimum number for which has a is the general vertex-distinguishing total chromatic number, denoted by .

It is evident that does exist for any graph . In this paper, we study the general vertex-distinguishing total coloring of some special classes of graphs and obtain the exact value of the general vertex-distinguishing total chromatic number of these graphs. Furthermore, we propose a conjecture on the upper bound of general vertex-distinguishing total chromatic number of a graph.

2. Main Results

We first present a trivial lower bound on the general vertex-distinguishing total chromatic number of a graph.

Theorem 3. Let be a graph on vertices. Then

Proof. Let . It follows that , so .

Notice that the lower bound of Theorem 3 can be attained in graphs, such as the -vertex path for . One can readily check that and for and for .

Theorem 4. Let be a graph without isolated vertices and isolated edges. Then

Proof. Suppose that is a - of . For any , let , where . Obviously, is a of .

We now turn to investigating the general vertex-distinguishing total chromatic number of an -vertex path.

Theorem 5. Let be a path on vertices, . Then

Proof. Denote by a path with vertex set and edge set . Let , and let be a - of . Let , , , and Evidently, , and (which implies ). In order to prove the conclusion, , it suffices to give a - of . When , it is not hard to construct the corresponding general vertex-distinguishing total colorings. Let . We first construct a - of recursively. Note that when , it has .
Procedure 1. Construct a 4- of (i.e., ) as follows: the vertices are colored by , respectively; the edges are colored by , respectively. It is easy to see that is a of .
Procedure 2. Construct a - of based on a - of . Let be
are colored by
when is even, are colored by
when is odd, are colored byIt should be pointed out that when is odd for , the colors’ form is with totally elements, and when is even for , the colors’ form is with elements in total.
According to , we can see that, for any , and , it follows that is not a total color set of vertices , , and . In addition, are as follows: (1 item), (2 items), (3 items),, ( items), and ( items). And . So, is a - of . We now show that also has a - based on a - of , for .
Let , and let be a - of constructed by Procedures 1 and 2. We first delete vertices from . Obviously, the resulting graph, denoted by , is isomorphic to . Let be for ; for ; and . Then is a - of .
All the above show that the conclusion holds.

According to Theorem 5, we have the same conclusion on cycles. Let be an -vertex cycle with vertex set and edge set .

Corollary 6. For any cycle , one has

Proof. Let and ; let also and let be a - of , constructed by the method of Theorem 5. Then we can extend to a - of by assigning color 1 to edge . So, the conclusion holds.

In the following (Theorem 7 to Theorem 9), we discuss the general vertex-distinguishing total chromatic number of some kinds of special trees. A star is the complete bipartite graph (). A double star is a tree containing exactly two vertices that are not leaves (which are necessarily adjacent). A tristar is a tree with vertex set and edge set , where are positive integers.

Theorem 7. For a star , one has

Proof. When , the conclusion is trivial. For , let , and . Since, for any ( is the vertex with ), for any - of , it follows that ; that is, . In order to prove , we need to show that there exists a - of . Otherwise, let be the graph with minimum such that does not have a -, where . Let be a vertex of degree 1 in . Consider the graph , obtained from by deleting the vertex and its incident edge. By the assumption of , has a -, denoted by . In addition, by interchanging the colors of some vertex and its incident edge appropriately, we can assume . Since , there is at least one set , for , which is not the total color set of the vertices of . So, on the basis of , in we can color and its incident edge by and , respectively. Obviously, the resulting coloring is a - of .

For two vertices of a graph , to identify these two vertices is to replace them by a single vertex (denoted by - in this paper) incident to all the edges which were incident in to either or . The resulting graph is denoted by . In what follows, we denoted by the set of .

Theorem 8. Let be a double star, and . Then

Proof. When , the results are easy to be proved. When , let be two vertices with degree more than 1, and . Evidently, the graph is isomorphic to the star . Let . Since for any , we have .
By Theorem 7, contains a - . Evidently, for any and . If , then we can extend to a - of by coloring vertices and edge with any three different colors in ; if , we without loss of generality assume . Let (resp., ) be the set of edges (except edge ) incident to (resp., ) in . We now extend to a - of as follows. By the fact that there remain vertices and edge uncolored in when is restricted to , we consider the following two cases. First, one of , say , satisfies that contains at most two elements. Assume ; we then color by , respectively, where when and when . The resulting coloring of is also denoted by . Then it follows in that , , and and . So, is a - of . Second, and ; then we will further discuss two subcases.(1)Consider . Suppose that (and ) is the set of vertices, except (or ), adjacent to (and ) in . Because is a - of , either or contains no vertices with total color set , for some . Without loss of generality we assume that there is no vertex with . For any vertex in such that and , interchange the two colors of and . The resulting coloring, still denoted by , satisfies that does not contain color . Then we color by any three colors in and obtain a - of .(2)Consider or ; assume here. Let . Color by and color by any two colors in . Obviously, the resulting coloring is a - of .
All the above show that . So, the conclusion holds.

Theorem 9. Let be a tristar defined as above, and . Then

Proof. When , the conclusion is easy to be checked; when or , since and , it follows that . In addition, it is not hard to give a 4- of in each case of or , so ; when , let . Identify vertices and in and let . By Theorem 8, has a - . With the analogous analysis method of Theorem 8, we can also extend to a - of . This shows . On the other hand, for any - of , it has that ; that is, . So, the result holds.

In the above, we construct a - of a graph by extending a - of graph , where is the resulting graph of identifying two vertices of degree more than 1 in . But this method does not always work. For instance, the graph shown in Figure 1(b) has a -, but the graph shown in Figure 1(a) does not contain any -. So any - of can not be extended to a - of .

In the following we are devoted to the study of the general vertex-distinguishing chromatic number of fan graph , wheel graph , and complete graph . Let be two graphs such that . The join of and is a graph with vertex set and edge set . A fan graph is defined as the join of a path of vertices and an isolated vertex. A wheel graph is defined as the join of a cycle of vertices and an isolated vertex.

Theorem 10. Let be a fan, ; then where .

Proof. Let and . When , the conclusion is easy to show. We now consider the case of (which implies ). Since and for , we can easily deduce that . So, it suffices to show that contains a -. In particular, we prove that contains a - such that the total color set of contains at least 5 elements. By induction on . When , it is not hard to construct such a - of . Suppose that, for any , , there exists a - of . Consider the fan graph , and let be a - of . Anyway, we can assume that, for any color , there is an edge for some such that (If not, we can permutate and ).
Note that for any edge of , , if we replace this edge by a vertex and connect this vertex to , , and , then the resulting graph, denoted by , is isomorphic to . We will use this to construct a - of based on . It is obvious that there remain only 4 uncolored elements, , and in , if we restrict to . We need to consider the following 2 cases.
Case 1. There exist colors such that . Let be the edge with , . In , let , , and , and the resulting coloring is still denoted by . Evidently, in , ; meanwhile and are the same as those in , and . So, a - of .
Case 2. Four different colors such that . Select an edge , , for which . Since is a - of , contains at least one element (here we assume ), say . Obviously, , , and in . We can permutate the colors so that and in , say . Then, in , erase the color of vertex and recolor it by color , and let , , , and . Obviously, in , it follows that , and are the same as those in , and . So, a - of .
All of the above show that has a -.

Theorem 11. Let be a wheel graph, ; then where .

We omit the proof for Theorem 11, since it is analogous to that of Theorem 10.

Theorem 12. For a complete graph , , one has

Proof. When the conclusion is easy to show. So we assume .
Denote by and the set of colors. For integer , we construct a - of as follows: let for any ; for , , and ; for ; for , , and ; ; ; . One can readily check that , for ; for ; ; and . Thus, is a - of , which shows .
Suppose that and is a - of . Since for any two vertices (), , one can see that there is at most one vertex whose total color set contains only one color. If there is a vertex with , without loss of generality assume ; then the total color set of each vertex contains color 1, which indicates If there is no vertex whose total color set contains only one color, then for each it has . Since, for any vertex , , it follows that and can not be two total color sets with respect to . This implies . So .
To prove , we need to show that has a -. In particular, we show that any has a - such that for each color there is at least one vertex in being colored by .
We prove this by induction on . When , 5- is the defined above for , where . Consider the graph obtained from by deleting vertex and its incident edges. Obviously, is isomorphic to . By the induction hypothesis, has a -, say , such that for each color there is at least one vertex of being colored by . Since , there must be some set (denoted by . We consider the following two cases.(1)Consider . By the induction hypothesis, each color appears at a vertex. Without loss of generality assume for . Then, is extended to a - of via coloring by one of the colors in ; coloring for by ; and coloring for by one of the colors in ().(2)Consider . Then on the basis of , we only need to color and all of its incident edges in by color .
One can readily check that the resulting coloring of in the above two cases is -s of such that each color in appears at a vertex of . Hence, has a -, and the conclusion holds.

In the following, we present a trivial upper bound of the general vertex-distinguishing total chromatic number of the join graph of two graphs.

Theorem 13. Suppose , are two simple graphs and . Then

Proof. Let = and . Suppose that is a - of and is a - of , where the sets of colors of and are and (), respectively.
Define as .
Combining colorings , we can obtain a - of .

3. Remarks

Based on the above results, we propose two conjectures as follows.

Conjecture 14. Let be a graph without isolated vertices. Then

Conjecture 15. Let be a connected graph on vertices. Then

Note that if Conjecture 15 is true, then Conjecture 14 is true. On the other hand, if Conjecture 14 is true, then the upper bound cannot be improved. For instance, the graph contains exactly three components. It is easy to show that .

In addition, there is a very interesting observation about the general vertex-distinguishing total chromatic number.

Observation 1. Let be a subgraph of a graph . Then it possibly follows that

As an illustration of this observation, we consider the path and the fan graph . is a subgraph of , while .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant nos. 61127005 and 61309015) and the National Basic Research Program of China (973 Program) (no. 2010CB328103).

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Copyright © 2014 Chanjuan Liu and Enqiang Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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