Graphs considered in this paper are finite, simple and undirected. We start with some basic notions and definitions of graphs. Let be a graph with and . For any vertex , the open neighborhood of is the set and the closed neighborhood is the set . Similarly, the open and closed neighborhoods of a set are respectively and . The degree of a vertex , denoted by , is the cardinality of its open neighborhood. The maximum and minimum degree of a graph is denoted by and , respectively. Given a set , we denote by the subgraph of induced by . Given any graph , a graph is -free if it does not have any induced subgraph isomorphic to . We denote by the path on vertices and by the cycle on vertices. A tree is a connected acyclic graph. The complete graph on vertices is denoted by and the complete graph of order 3 is called a triangle. The complete bipartite graph with classes of orders and is denoted by . A star is a graph with . A bi-star is a graph formed by two stars by adding an edge between the center vertices. For any undefined terms, the reader is referred to the book of Bondy and Murty Bondy and Murty (2008).
A proper vertex -coloring of a graph is a mapping such that any two adjacent vertices receive different colors. In fact, this problem is equivalent to the problem of partitioning the vertex set of into independent sets where . The set of all vertices colored with the same color is called a color class. The chromatic number of , denoted by , is the minimum number of colors among all proper colorings of . A graph admitting a proper -coloring is said to be -colorable, and it is said to be -chromatic if its chromatic number is exactly .
A dominating set is a subset of the vertices in a graph such that every vertex in either belongs to or has a neighbor in . The domination number is the minimum cardinality of a dominating set of . A -set is a dominating set of with minimum cardinality.
Graph coloring and domination are two major areas in graph theory and both have been well studied. There exist plenty of variants of classical graph coloring Chen and Schelp (1998); Malaguti and Toth (2010). Also, excellent surveys on the fundamentals of domination and several advanced topics are given in Haynes et al. (1998) and Haynes et al. (1998), respectively. Moreover, graph coloring and domination problems are often in relation. Chellali and Volkmann Chellali and Volkmann (2004) showed some relations between the chromatic number and some domination parameters in the graph. Indeed, Hedetniemi et al. Hedetniemi et al. (2009) introduced the concepts of dominator partition of a graph. Motivated by Hedetniemi et al. (2009), Gera et al. Gera et al. (2006) proposed the dominator coloring as a proper coloring such that every vertex has to dominate at least one color class (possibly its own class) in 2006. The minimum number of colors among all dominator colorings of is the dominator chromatic number of , denoted by . Gera studied further this coloring problem in Gera (2007a, b). More results on the dominator coloring could be found in Chellali and Maffray (2012); Merouane and Chellali (2012); Arumugam et al. (2012); Guillaume et al. (2017). In 2015, Boumediene et al. Merouane et al. (2015) introduced the dominated coloring as a proper coloring where every color class is dominated by at least one vertex. The minimum number of colors among all dominated colorings of is the dominated chromatic number of , denoted by .
For problems mentioned above, the domination property is defined either on vertices or on color classes. Indeed, the color classes in a dominator coloring are not necessarily all dominated by a vertex, and the vertices in a dominated coloring are not necessarily all dominates a color class. In this paper, we introduce the domination coloring that both of the vertices and color classes should satisfy the domination property. A domination coloring of a graph is a proper vertex coloring of such that each vertex of dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all domination colorings, denoted by , is called the domination chromatic number.
We concern with connected graphs only. The aim of this paper is to study properties and realizations of the dominator chromatic number. In Section 2, we analyse the basic complexity of the domination coloring parblem. In Section 3, we present the domination chromatic number for classes of graphs, and find bounds and characterization results. We investigate some realization results in Section 4, and pose open questions in Section 5.
2 Basic complexity results
This section focuses on the complexity study of the domination coloring problem. Whether an arbitrary graph admits a domination coloring with at most colors? We aim to this decision problem and give the following formalization:
-Domination Coloring Problem:
Instance: A graph without isolated vertices and a positive integer .
Question: Is there a domination coloring of with at most colors?
For , the -domination coloring problem is NP-Complete.
Proof. The -Domination Coloring Problem is in NP, since verifying if a coloring is a domination coloring could be performed in polynomial time. Now, we give a polynomial time reduction from the -Coloring Problem which is known to be NP-Complete, for . Let be a graph without isolated vertices. We construct a graph from by adding a new vertex to and adding edges between and every vertex of . That is, is a dominating vertex of , as shown in Figure 1. We show that admits a proper coloring with colors if and only if admits a domination coloring with colors.
First, we prove the necessity. Let be a proper -coloring of , and the corresponding color classes set is . We construct a -domination coloring of with the color classes set . It is easy to see that is a domination coloring of since
Each vertex other than dominate at least the color class containing and dominate all color classes of ;
Each color class other than is dominated by and the color class containing is dominated by any other vertex.
Then, we prove the sufficiency. Let be a - domination coloring of , and is the color classes set. Since is proper, there exists a color class such that . So, we can construct a proper -coloring of by removing the color class from .
From the above, the -Domination Coloring Problem is NP-Complete, for .
3 Domination coloring for classes of graphs
In this section, we study some properties of the domination coloring and basic results on typical classes of graphs.
Let be a connected graph with order . Then at least two different colors are needed in a domination coloring since there are at least two vertices in adjacent to each other. Moreover, if each vertex receives a unique color, then both the vertices and color classes satisfy the domination property. Clearly, we get a domination coloring of with colors. Thus,
Gera et al. Gera et al. (2006) introduced the Inequalities (2) for the dominator chromatic number and Boumediene et al. Merouane et al. (2015) obtained Inequalities (3) for the dominated chromatic number . Also, we can get a similar inequality for the domination chromatic number .
Let be a graph without isolated vertices, then
Proof. It is easy to check the left two parts of the inequality. Consider a graph and a -set of . We obtain a domination coloring of by giving distinct colors to each vertex of and at most new colors to the vertices of . Hence, we totally use at most colors. So, .
Let be a planar graph without isolated vertices, then .
(1) For the path , ,
(2) For the cycle ,
(3) For the complete graph , ;
(4) For the complete -partite graph , ;
(5) For the complete bipartite graph , ;
(6) For the star , ;
(7) For the wheel ,
Proof. (1) Let . By the definition of domination coloring, we discover that at most two non-adjacent vertices are allowed in a color class, if not, there exist no vertex dominating this color class. On the other hand, the vertex adjacent to both of the two vertices of a color class must be the unique vertex of some color class. For convenience, let be a -subgraph of . If vertices and are in a color class, then must be the unique vertex of a color class. If not, and are partitioned in a color class, which will result in cannot dominate any color class. So, every three vertices of need be partitioned in two color classes, and each of the rest vertices forms its own color class. Clearly, it is an optimal domination coloring of . Thus, .
(2) For , the result follows by inspection. For , it is not hard to find the case is similar to the path . As the discussion in (1), the result follows.
(4) Let be the complete -partite graph, and be the -partite sets. Then . Also, the coloring that assigns color to each partite set is a domination coloring. The result follows.
(5) and (6) are special cases of (4).
(7) Let be the wheel with order . Since,
and the corresponding proper colorings are also domination colorings, the result follows.
One note on the domination chromatic number is that for a given graph , and a subgraph of , the domination chromatic number of can be smaller or larger than the domination chromatic number of . That is to say, induction may be not useful when we want to find the domination chromatic number of a graph. As an example, consider the graph and , then , and consider the graph and , then .
Let be a connected graph with order . Then if and only if for .
Proof. By Theorem 1 (5), if , then . We just need to prove the necessity.
Let be a connected graph such that , and and are the two color classes. If or , then . So, suppose that and . For any vertex , since , it follows that dominates color class . Similarly for any vertex in . Thus, each vertex of is adjacent to each vertex of , and both and are independent. So for , and the result follows.
Let be a connected graph with order . Then if and only if for .
Proof. By Theorem 2 (3), , if . We only need to prove the necessity.
Let be a connected graph with . Suppose that . Thus, there exist at least two vertices, say and , such that they are not adjacent in . Now, we define a coloring of in which and receive the same color, and each of the remaining vertices receive a unique color. This is a domination coloring, so , a contradiction. Thus, , and we obtain the result.
Next, we consider the bi-stars. Let be the bi-star with central vertices and , where and . Let and . Obviously, and , as shown in Figure 2.
For the bi-star with , .
Proof. Consider a proper coloring of in which the color classes , , , and . Then, each vertex in the set dominates the color class , and each vertex in the set dominates the color class . Also, the color class is dominated by any vertex in , is dominated by any vertex in , is dominated by vertex , and is dominated by vertex . Therefore, this is a domination coloring, and .
By the Lemma 2.2 in Gera et al. (2006), . So, . Suppose that . It will be result in that each vertex in or each vertex in does not dominate a color class. Thus, .
For the Petersen graph , .
Let be a connected graph with order and maximum degree , then .
Proof. Consider a minimum domination coloring of . Since is -free, any color class would not have more than vertices; otherwise, a vertex dominating such color class will induce a star of order at least , a contradiction. So, .
Let be a connected triangle-free graph, then .
Proof. Consider a minimum dominating set of . Color every vertex of with a new color. Since does not contain any triangle, the set of neighbors of every vertex of is an independent set. So, a second new color is given for each neighborhood. Obviously, this is a proper coloring of with colors, which satisfies that every vertex dominates at least one color class, and every color class is dominated by at least one vertex. Thus, .
4 Graphs with
For any graph , we have . In this section, we investigate graphs for which .
Let be a connected graph, if , then .
Proof. Let be a connected graph of order with . It then follows that is a spanning subgraph of . Let be a vertex of degree in . Since is adjacent to all the vertices of , a minimum coloring of uses colors. Also, this coloring is also a domination coloring of , where each vertex dominates the color class of , the color class is dominated by any other vertex and each of other color classes is dominated by . So, , and then .
A unicyclic graph is a graph which contains only one cycle. In the following， we characterize unicyclic graphs with .
Let be a connected unicyclic graph. Then if and only if is isomorphic to or or or the graph obtained from by attaching any number of leaves at one vertex of .
Proof. For the sufficiency, the result is obvious if is the graph meet conditions. We consider only the necessity. Let be a connected unicyclic graph with , and the unique cycle of .
Case 1. If is an even cycle, then and . It follows that cannot contain any other vertices not on , otherwise . By Theorem 2(2), .
Case 2. If is an odd cycle, then . Suppose there exists a support vertex not on . Since or the leaf is a color class in each -coloring of , it follows that , which is a contradiction. Hence, all the support vertices lie on , and any vertex not on is a leaf. Morever, the number of support vertices is at most one. Otherwise, it follows that some color class does not be dominated, since there exists some -coloring of in which every support vertex appears as a singleton color class.
Case 2.1. If , then is isomorphic to or the graph obtained from by attaching any number of leaves at exactly one vertex of .
Case 2.2. Suppose that . If there exists a support vertex on , then there exists a -coloring of such that contains all the leaves of . Now, we get two vertices and on , such that , both and are not adjacent to . Clearly, does not dominate any color class and the color class does not be dominated by any vertex, which is a contradiction. Thus, has no support vertices and . By Theorem 2(2), . So, the theorem follows.
For the complete graph , we know that . Next, we construct a family of graphs by attaching leaves at some vertices of the complete graph. We denote the family of graphs obtained by attaching leaves at vertices of , . We take no account of the number of leaves attached at any vertex in the notation, since it does not impact the domination chromatic number. Morever, we denote any element in by . For example, a instance of is shown in Figure 4.
For , .
Proof. For any , is -colorable. So, . Next, we consider a domination coloring of . On one hand, each vertex attached leaves should be partitioned into a singleton color class, since each of these leaves has to dominate a color class formed by its only neighbor. On the other hand, leaves attached to different vertices have to be partitioned into different color classes, otherwise, there exist no vertex dominating the color class. Thus, at most vertices can be attached leaves of , in order to guarantee that is -domination colorable. The result follows.
5 Conclusion and Further Research
In this paper, we propose the concept of domination coloring based on the dominator coloring and dominated coloring. Basic results and properties of the domination coloring number are studied. Also, we characterize two classes of graph that have the property that .
The following are some interesting problems for further investigation.
For what graphs does ?
For what graphs does ?
For what graphs does ?
The authors acknowledge support from Peking University. The authors also would like to thank referees for their useful suggestions.
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