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Star Coloring and Tree-Width of the Kneser Graph KG(N,2)

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Star Coloring and Tree-Width of the Kneser Graph KG(N,2) Star Coloring and Tree-width of the Kneser Graph KG(n; 2) Behnaz Omoomi and Elham Roshanbin Department of Mathematical Sciences Isfahan University of Technology 84156-83111, Isfahan, Iran Abstract A proper coloring of the vertices of a graph G is called a star coloring if any path of length three in G is not 2-colored. The star chromatic number of G is the minimum number of colors required to obtain a star coloring of G. In this paper, we give the exact value of the star chromatic number of the Kneser graph KG(n; 2). Moreover, we obtain a lower bound and an upper bound for the tree-width of these graphs. Keywords: Star coloring; Tree-width; Kneser graph. 1 Introduction Throughout this paper, all graphs are finite and simple. We use u ∼ v and u ≁ v to respectively denote the adjacency and non-adjacency relations between vertices u and v. We denote a path and a cycle on n vertices by Pn and Cn, respectively. We refer the reader to [12] for graph-theoretic notation and terminology not described in this paper. A proper vertex coloring of a graph G is an assignment of colors to the vertices of G such that no two adjacent vertices receive the same color. Given a proper coloring c of graph G and any subset A ⊆ V (G), we let c(A) denote the set of all colors used to color vertices in A. A proper vertex coloring of a graph G is called a star coloring if any path of length three in G is not 2-colored; equivalently, the union of every two color classes in G induces a forest whose components are stars. The star chromatic number of 1 G, denoted by χs(G), is the minimum number of colors required to obtain a star coloring of G. Star colorings of graphs were introduced by Gr¨unbaum in 1973 [4] (see also [1, 3, 9]). The Kneser graph KG(n; k), n ≥ 2k, is a graph whose vertices are the k-element subsets of an n-element set, where two vertices are adjacent if and only if the two corresponding sets are disjoint. In this paper, vertices of KG(n; 2) are the 2-element subsets fi; jg, 1 ≤ i < j ≤ n, which we denote ij. Kneser graphs have many interesting properties and have been the sub- ject of much research. It was conjectured by Kneser in 1955 [7] and proved by Lov´aszin 1978 [8] that χ(KG(n; k)) = n − 2k + 2. Since then several types of colorings of Kneser graphs have been considered. For example, the circular chromatic number, the b-chromatic number and the multi- chromatic number of Kneser graphs were investigated in [5], [6] and [11], respectively. The concept of tree-width of a graph introduced by Robertson and Seymour in 1984 [10] to( measure) how tree-like a graph behaves. Fertin et t+2 al. [3] gave the bound 2 on the star chromatic number of graphs with tree-width t. In this paper, we obtain the exact value of the star chromatic number of the Kneser graph KG(n; 2), for n ≥ 5. We also give a lower bound and an upper bound on the tree-width of KG(n; 2). 2 Main results ( ) n−1 ≥ In this section, we show that χs(KG(n; 2)) = 2 , for n 6 and χs(KG(5; 2)) = 5. First, we prove the following proposition. Proposition 1. Let G = (X; Y ) be a bipartite graph, such that jXj = jY j = n, and each vertex in G has degree n − 1. Then, χs(G) = n. Moreover, there are only two types of optimum star colorings of G. Proof. Let c be a star coloring of G. If the vertices of X or Y all receive different colors, then the number of colors used in c is at least n. Otherwise, there are vertices u1; u2 2 X and v1; v2 2 Y , where c(u1) = c(u2) and c(v1) = c(v2). Since there is no 2-colored P4 and G is (n − 1)- regular, neither u1 nor u2 is adjacent to both vertices v1 and v2. Say 2 u1 ≁ v1 and u2 ≁ v2. Therefore, there are no other pairs of vertices with the same color in the same part (except u1; u2 and v1; v2), because all vertices in Y (similarly in X), except v1 (u1), are adjacent to u1 (v1), and (G; c) does not have a 2-colored P4. Hence, in such a star coloring of G, at least 2 + (n − 2) = n colors are needed. Thus, χs(G) ≥ n. In Figure 1, two possible types of optimum star colorings of G with n colors are shown. Hence, χs(G) = n. ■ Figure 1: Two optimum star colorings of G. Note that, for every n ≥ 6, the vertex set of the Kneser graph KG(n; 2) can be decomposed into three subsets f12g, A and B; where A and B are the sets of all vertices adjacent( and) non-adjacent to vertex 12, respectively. j j n−2 It is easy to see that, A = 2 , and the induced subgraph on B is a bipartite graph with parts X := f1i : 3 ≤ i ≤ ng and Y := f2i : 3 ≤ i ≤ ng. By Proposition 1, χs(B) = n − 2. Thus, we can give a color assignment to the vertices of B, the same as the first( star) coloring given in the proposi- n−2 tion above (Figure 1). Then, we assign 2 new colors to the vertices of A, and, finally, give one of the colors of the vertices of B, to the vertex 12. It can( be) easily checked( that,) this coloring is a star coloring( of KG) (n; 2) n−2 − n−1 ≤ n−1 with 2 + n 2 = (2 ) colors. Thus, χs(KG(n; 2)) 2 . Fur- n ther, since KG(n; 2) has 2 vertices, in a given optimum star coloring c of KG(n; 2), there are at least two vertices with a same color. Without loss of generality (after a renaming if necessary), suppose that c(13) = c(23) = a. To prove the main result of this section, we need the following lemmas, which will make use of the above notation. Specifically, A, B, X, Y , and c continue to be used throughout this section. Lemma 1. Let A0 := fkl : 4 ≤ k < l ≤ ng together with A00 := f3l : 4 ≤ l ≤ ng be a partition of A. In the optimum star coloring c of graph KG(n; 2) we have (I) c(B) \ c(A0) = ;. 3 ( ) j 0 j n−3 (II) c(A ) = 2 . Proof. (I) The color of any vertex in X [ Y can not be the same as any 0 vertex in A , otherwise there would be a 2-colored P4 containing 13 and 23, which is a contradiction. (II) Since all vertices in A0 are adjacent to 13 and 23, there are no two 0 vertices in A with the same color, otherwise there would be a 2-colored P4 containing 13 and 23, which contradicts that c is a star coloring. ■ Let t be the number of pairs (1i; 2i), i ≥ 4, of vertices in B, such that c(1i) = c(2i). Then we have the following lemma. Lemma 2. If t ≥ 1, or equivalently there is some i, 4 ≤ i ≤ n, such that c(1i) = c(2i), then we have the following facts. (I) All vertices in A00nf3ig must have different colors. In other words, only for one l ≥ 4, l =6 i, it is possible that c(3l) = c(3i), i.e., a repeated color in A00. (II) If c(A0) \ c(A00) =6 ;, then c(3l) = c(il) or c(3i) = c(li), for some l ≥ 4. (III) If c(A00) \ c(B) =6 ;, then either c(3i) = c(1i) = c(2i) or c(3i) = c(13) = c(23) = a. (IV) If c(A0) \ c(A00) =6 ; and there exists l ≥ 4, such that c(3l) = c(il), then c(12) 62 fa; c(1i)g. Proof. (I) Otherwise, there are k; k0 =6 i (noting that n ≥ 6), such that c(3k) = c(3k0) and the path 3k 1i 3k0 2i is 2-colored. (II) Otherwise, if there exists k and l, with l; k =6 i and c(3l) = c(kl), then the path 3l 1i kl 2i is 2-colored. (III) Otherwise, suppose that there is l =6 i, and c(3l) = c(1l) (similarly c(3l) = c(2l)). Then 1i 3l 2i 1l is 2-colored (2i 3l 1i 2l is 2-colored). If there exists l =6 i, and c(3l) = c(13) = c(23) = a, then the path 13 2i 3l 1i is 2-colored. (IV) If there exists l ≥ 4, such that c(3l) = c(il) and c(12) 2 fa; c(1i)g, then the path 13 il 12 3l or the path 1i 3l 12 il is 2-colored. ■ Now we are ready to prove our main theorem, the proof of which uses the above notation. 4 ( ) ≥ n−1 Theorem 1. For n 6, χs(KG(n; 2)) = 2 , and χs(KG(5; 2)) = 5. ( ) ≥ ≤ n−1 Proof. Let n 6. We have already shown that χs(KG(n; 2)) 2 . To prove the equality we need( to show) that every (optimum) star coloring n−1 of KG(n; 2) requires at least 2 colors. We consider the following two possibilities. Case 1: In the optimum star coloring c, there are no two vertices with a same color in the same part of B.
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