Total dominator chromatic number of Kneser graphs

In this paper among some other results and by using the existance of Steiner triple systems, we determine the total dominator chromatic number of the Kneser graph KG(n,2).


Introduction
Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures, Let G = (V, E) be a graph with the vertex set V of order n(G) and the edge set E of size m(G). The open neighborhood and the closed neighborhood of a vertex v ∈ V are N G (v) = {u ∈ V | uv ∈ E} and N G [v] = N G (v) ∪ {v}, respectively. The degree of a vertex v is also deg G (v) = |N G (v)|. An isolated vertex is a vertex with degree zero. A proper coloring of a graph G = (V, E) is a function from the vertices of the graph to a set of colors such that any two adjacent vertices receive different colors. The chromatic number χ(G) of G is the minimum number of colors needed in a proper coloring of a graph. In a proper coloring of a graph, a color class of the coloring is a set consisting of all those vertices with the same color. If f is a proper coloring of G with the coloring classes V 1 , V 2 , . . . , V ℓ such that every vertex in V i has color i, we write simply f = (V 1 , V 2 , . . . , V ℓ ). A total dominating set, briefly T DS, S of a graph G [4] is a subset of the vertices in G such that for each vertex v, N G (v) ∩ S = ∅. The total domination number γ t (G) of G is the minimum cardinality of a T DS of G. Motivated by the relation between coloring and domination, the notion of total dominator coloring was introduced in [7]. Also the reader can consult [2,3,8] for more information. A total dominator coloring, briefly TDC, of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number χ td (G) of G is the minimum number of color classes in a TDC of G. A nice relation between these graphical invariants is provided in [3] as follows.
Proposition 1.1 [3] For each isolated free graph G we have For positive integers n, k with the condition k ≤ n 2 let V = [n] k be the set of all k−subsets of the nset {1, 2, . . . , n}. The Kneser graph KG(n, k), is a graph with the vertex set V such that two vertices are adjacent in it if and only if the corresponding subsets are disjoint. For example KG(2k, k) is a matching and KG(5, 2) is the Petersen graph. Note that each vertex in KG(n, 2) which is a 2−subset of the set {1, 2, . . . , n} corresponds to an edge in the complete graph K n with vertex set {1, 2, . . . , n}. Hence, two vertices of KG(n, 2) are non-adjacent if and only if the corresponding edges in K n are adjacent. Also, If A is an independent set of vertices in KG(n, 2), then either all vertices in A have a common symbol , say a, or A = {ab, ac, bc}, for some a, b, c ∈ {1, 2, . . . n}. In other word, an independent set of vertices in KG(n, 2) corresponds to a star subgraph with center a (and in this case "a" is considered to be a central symbol) or a triangle subgraph in K n . From now on, we call the independent set (color class) in KG(n, 2) of the first form starlike with center a and the second form triangular. Moreover, for convenient we denote the vertex {a, b} with ab. When a starlike color class consists of only one vertex, say ij, then we choose in arbitray exactly one of these two symbols i or j as the central symbol of this color class. Note that two different starlike color classes may have the same central symbols. Since every proper coloring is a partition of vertices into independent sets of vertices, we can consider every proper coloring of KG(n, 2) as an edge decomposition of the complete graph K n into star and triangle subgraphs. The total domination number of the Kneser graphs KG(n, 2) is completely determned in [5] as follows. Also, it is well known that χ(KG(n, k)) = n − 2k + 2 and hence χ(KG(n, 2)) = n − 2, see [10]. In recent years many different types of colorings for the Kneser graphs has been studied by several researchers. For instance, the b−chromatic number of some Kneser graphs is investigated in [6] and the locating chromatic number of Kneser graphs is studied in [1] in which the locating chromatic number of KG(n, 2) is completely determined. In this paper, we determine the total dominator coloring of the Kneser graph KG(n, 2).
Lemma 2.2 Let k ≥ n − 2 and f = (V 1 , V 2 , . . . , V k ) be a proper coloring of the Kneser graph KG(n, 2). If there exists a symbol i ∈ {1, 2, ..., n} such that "i" appears in at least k − 1 color classes, then f is not a TDC.
Proof. Since for each i ∈ {1, 2, ..., n} we have |{ij : i = j, 1 ≤ j ≤ n}| = n − 1, the number of vertices in KG(n, 2) which contain the symbol "i" is n − 1. Hence, each symbol can appear in at most n − 1 color classes. First assume that "i" appears in all of the color classes and hence, k ≤ n − 1. Let j ∈ {1, 2, . . . , n} \ {i} and note that in each color class there exists (at least) one vertex which contains the symbol i and hence, is not adjacent to the vertex ij. This means that the vertex ij can not be adjacent to all of the vertices of a color class and thus f is not a T DC. Now, suppose that the symbol i appears in exactly k − 1 color classes and there exist a color class V j such that i does not appear in it. Let i ′ j ′ be a vertex in V j . Note that in this case the vertex ii ′ can not be adjacent to all of vertices of a color class which means that f is not a TDC and this completes the proof.
Now the following result is obtained.
Proof. By Theorem 2.3, in each proper (n − 2)−coloring of the Kneser graph KG(n, 2), there exists a symbol (namely n, n − 1 or n − 2 by renaming the symbols if it is necessary) which appears in all of the color classes. Thus, by using Lemma 2.2 the result follows. Proof. Since there are n − 1 starlike color classes, there exist at most n − 1 (distinct) central symbols. Hence, there exists a non-central symbol i ∈ {1, 2, ..., n}. Therefor, the symbol i appears at most once in each color class. Since the number of vertices containing the symbol i is n − 1, this symbol should appear exactly once in each of the n − 1 color classes. Now Lemma 2.2 implies that f is not a T DC.

Proposition 2.6
The total dominator chromatic number of the Petersen graph is 6 and for each interger n ≥ 6 we have χ td (KG(n, 2)) ≤ n.
Proof. It is not hard to see by investigation that there does not exist a T DC for the Petersen graph with 3,4 or 5 color classes and in Table 1 a T DC of KG(5, 2) with 6 color classes is presented. Hence, χ td (KG(5, 2)) = 6. Also, it is easy to check that for each n ∈ {5, 6, 7, 8, 9} the corresponding vertex partition of KG(n, 2) presented in Table 1 provides a total dominator coloring for KG(n, 2). In fact the general model for color classes for each n ≥ 7 can be stated as follows. Assume that n ≥ 7. Let This partition of the vertices of KG(n, 2) provides a proper n-coloring for KG(n, 2) in which the color class V 3 is triangular and all other color classes are starlike. Note that each vertex in V 1 ∪ V 2 ∪ · · · ∪ V 6 is adjacent to all of vertices of some V j , j ∈ {1, 2, ..., 6}. Also, for each i ≥ 7 the vertex ij is adjacent to at least one of the vertices 12 or 34 and this means that the vertex ij is adjacent to all of vertices of V 1 or V 2 . Therefor, this vertex partition provides a T DC for KG(n, 2) and hence χ td (KG(n, 2)) ≤ n for each n ≥ 7.  Proof. Lemma 2.5 implies that t ≥ 1. Let ℓ be the number of starlike color classes and hence,

KG
By renaming the symbols 1, 2, . . . , n, if it is necessary, we may assume that the centers of these ℓ starlike classes (which some of them may have similar centers) are included in the ℓ-set {n, n − 1, . . . , n − (ℓ − 1)}, and hence we can asuume that these starlike color classes are indexed as V n , V n−1 , . . . , V n−ℓ+1 . Therefore, n − ℓ symbols 1, 2, . . . , n − ℓ are not the central symbol of any starlike color class. The number of vertices of KG(n, 2) whose both symbols lie in the set {1, 2, . . . , n−ℓ} is equal to n−ℓ 2 and all of these vertices must be distributed among triangular color classes. Since each triangular color class contains exactly three vertives, for the distribution of these n−ℓ 2 vertices at least 1 3 n−ℓ 2 triangular color classes is needed. Thus, we must have and the relation t = (n − 1) − ℓ implies that Since n − ℓ − 1 = 0, we conclude that t ≤ 5.
Recall that a balanced incomplete block design (a BIBD) with parameters t, n, k, λ (i.e. a t−(n, k, λ) design) is an ordered pair (S, β) in which S is a set of n points (or symbols) and β is a family of k−subsets of S called blocks, such that every t elements of S occur together in exactly λ blocks of β. When λ = 1 the design is called a Steiner system, and when k = 3 it is called a triple system. A design with parameters t = 2, k = 3 and λ = 1 with n points is called a Steiner triple system of order n, denoted by ST S(n). It is well known that a Steiner triple system of order n exists if and only if n ≡ 1, 3 (mod 6), see [9]. Now we are ready to state the main theorem as follows. Proof. By using Corolary 2.4 and Proposition 2.6, χ td (KG(5, 2)) = 6 and for each n ≥ 6 we have n − 1 ≤ χ td (KG(n, 2)) ≤ n. Here after we assume that n ≥ 6 and we want to show that χ td (KG(n, 2)) = n − 1. Suppose on the contrary that there exists a total dominator coloring of KG(n, 2) with n − 1 color classes. Thus, by Lemma 2.7 and by using the notations appeared in it and in its proof, we have 1 ≤ t ≤ 5. Let C ⊆ {1, 2, ..., n} denotes the set of central symbols and let C = {1, 2, ..., n} \ C. By considering the following cases based on different possibilities for t, we show that each of them leads to a contradiction, and this completes the proof.
Since each V s , 1 ≤ s ≤ 4, is an independent set in triangular form and there exist two different vertices x = ij and y = i ′ j ′ in 4 s=1 V s such that Now we consider the following two subcases. Subcase 2.1. There exists 1 ≤ s ≤ 4 such that {x, y} ⊆ V s . Without loss of generality, assume that V s = {x, y, 12} and x = 1j, y = 2j (note that V s is a triangular color class and hence j = j ′ ). "Three" vertices 13, 14, 15 should be distributed among the three remaining triangular color classes. We know that each triangular color class which contains a vertex containing the symbol "1", should contain exactly two vertex containing the symbol 1. Thus, an even number of vertices in ( Subcase 2.2. x ∈ V s and y ∈ V s ′ with s = s ′ and 1 ≤ s, s ′ ≤ 4. In this case, V s should contain two vertices containing the symbol j and V s ′ should contain two vertices containing the symbol j ′ . This means that which is a contradiction.
If k ≥ 5, then k 2 vertices which both of their symbols are non-central, should be distributed among 3 triangular color classes which is impossible since k 2 ≥ 5 2 = 10 and there are 9 vertices in these 3 triangular color classes. Thus we have exactly 4 non-central symbols, say C = {1, 2, 3, 4}. Let V 1 , V 2 , V 3 be these triangular color classes. Since We consider three subcases as below.
Then, V t should contain two vertices containing the symbol j ′′ . This means that which is a contradiction.
Similarly, V 1 should contain two vertices containing the symbol j, V 2 should contain two vertices containing the symbol j ′ and V 3 should contain two vertices containing the symbol j ′′ . This means that which is a contradiction.
Case 5: Let t = 1. Let V 1 be the triangular color classes. Since ℓ = n − 2 and |C| ≤ n − 2, there are k ≥ 2 non-central symbols. Assume that {1, 2} ⊆ C and hence V 1 contains the vertex 12. Without lose of generality, assume that V 1 = {12, 13, 23}. Let i ∈ {4, 5, ..., n}. Since the number of vertices containing the symbol "i" is n − 1 and there are n − 2 (starlike) color classes which may contain a vertex with symbol i, by the Pigenhole principle there exists a starlike color class with at least two vertices containing the symbol i, i.e a starlike with central symbol i. Hence, {4, 5, ..., n} ⊆ C and for each i ∈ {4, 5, ..., n} the vertex 1i should be in a starlike color class with central symbol i. Therefore, the symbol 1 appears in at least n − 2 color classes and this contradicts Lemma 2.2.
The authors declare that they have no competing interests.