More Bounds for the Grundy Number of Graphs

More bounds for the Grundy number of graphs∗ Zixing Tang, Baoyindureng Wu †, Lin Hu College of Mathematics and System Sciences, Xinjiang University Urumqi, Xinjiang 830046, P.R.China Manoucheher Zaker ‡ Department of Mathematics, Institute for Advanced Studies in Basic Sciences, 45137-66731 Zanjan, Iran Abstract A coloring of a graph G = (V, E) is a partition V ,V ,...,Vk of V { 1 2 } into independent sets or color classes. A vertex v Vi is a Grundy vertex if it is ∈ adjacent to at least one vertex in each color class Vj for every j < i. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the Grundy number Γ(G) of a graph G is the maximum number of colors in a Grundy coloring. We provide two new upper bounds on Grundy number of a graph and a stronger version of the well-known Nordhaus-Gaddum theorem. In addition, we give a new character- ization for a P ,C -free graph by supporting a conjecture of Zaker, which says { 4 4} that Γ(G) δ(G) + 1 for any C -free graph G. ≥ 4 Keywords: Grundy number; Chromatic number; Clique number; Coloring number; Randićindex 1 Introduction All graphs considered in this paper are finite and simple. Let G = (V, E) bea finite simple graph with vertex set V and edge set E, V and E are its order and | | | | size, respectively. As usual, δ(G) and ∆(G) denote the minimum degree and the maximum degree of G, respectively. A set S V is called a clique of G if any two arXiv:1507.01080v2 [math.CO] 8 Dec 2015 ⊆ vertices of S are adjacent in G. Moreover, S is maximal if there exists no clique properly contain it. The clique number of G, denoted by ω(G), is the cardinality of a maximum clique of G. Conversely, S is called an independent set of G if no two vertices of S are adjacent in G. The independence number of G, denoted by α(G), is the cardinality of a maximum independent set of G. The induced subgraph ∗Research supported by NSFC (No. 11161046) and by Xinjiang Talent Youth Project (No. 2013721012) †Corresponding author. Email: [email protected] (B. Wu) ‡Email: [email protected] (M. Zaker) 1 of G induced by S, denoted by G[S], is subgraph of G whose vertex set is S and whose edge set consists of all edges of G which have both ends in S. Let C be a set of k colors. A k-coloring of G is a mapping c : V C, → such that c(u) = c(v) for any adjacent vertices u and v in G. The chromatic 6 number of G, denoted by χ(G), is the minimum integer k for which G has a k- coloring. Alternately, a k-coloring may be viewed as a partition V ,...,Vk of V { 1 } into independent sets, where Vi is the set of vertices assigned color i. The sets Vi are called the color classes of the coloring. The coloring number col(G) of a graph G is the least integer k such that G has a vertex ordering in which each vertex is preceded by fewer than k of its neighbors. The degeneracy of G, denoted by deg(G), is defined as deg(G)= max δ(H) : H { ⊆ G . It is well known (see Page 8 in [17]) that for any graph G, } col(G)= deg(G) + 1. (1) It is clear that for a graph G, ω(G) χ(G) col(G) ∆(G) + 1. (2) ≤ ≤ ≤ Let c be a k-coloring of G with the color classes V ,...,Vk . A vertex v Vi { 1 } ∈ is called a Grundy vertex if it has a neighbor in each Vj with j < i. Moreover, c is called a Grundy k-coloring of G if each vertex v of G is a Grundy vertex. The Grundy number Γ(G) is the largest integer k, for which there exists a Grundy k-coloring for G. It is clear that for any graph G, χ(G) Γ(G) ∆(G) + 1. (3) ≤ ≤ A stronger upper bound for Grundy number in terms of vertex degree was obtained in [32] as follows. For any graph G and u V (G) we denote v V (G) : ∈ { ∈ uv E(G), d(v) d(u) by N (u), where d(v) is the degree of v. We define ∈ ≤ } ≤ ∆ (G) = max max d(v). It was proved in [32] that Γ(G) ∆ (G)+1. 2 u∈V (G) v∈N≤(u) ≤ 2 Since ∆ (G) ∆(G), we have 2 ≤ Γ(G) ∆ (G) + 1 ∆(G) + 1. (4) ≤ 2 ≤ The study of Grundy number dates back to 1930s when Grundy [16] used it to study kernels of directed graphs. The Grundy number was first named and studied by Christen and Selkow [7] in 1979. Zaker [29] proved that determining the Grundy number of the complement of a bipartite graph is NP-hard. A complete k-coloring c of a graph G is a k-coloring of the graph such that for each pair of different colors there are adjacent vertices with these colors. The achromaic number of G, denoted by ψ(G), is the maximum number k for which the graph has a complete k-coloring. It is trivial to see that for any graph G, Γ(G) ψ(G). (5) ≤ 2 Note that col(G) and Γ(G) (or ψ(G)) is not comparable for a general graph G. For instacne, col(C4) = 3, Γ(C4)=2= ψ(C4), while col(P4) = 2, Γ(P4) = 3= ψ(P4). Grundy number was also studied under the name of first-fit chromatic number, see [18] for instance. 2 New bounds on Grundy number In this section, we give two upper bounds on the Grundy number of a graph in terms of its Randićindex, and the order and clique number, respectively. 2.1 Randićindex The Randićindex R(G) of a (molecular) graph G was introduced by Randić [26] in 1975 as the sum of 1 over all edges uv of G, where d(u) denotes the √d(u)d(v) degree of a vertex u in G, i.e, R(G) = P 1/pd(u)d(v). This index is quite uv∈E(G) useful in mathematical chemistry and has been extensively studied, see [19]. For some recent results on Randićindex, we refer to [10, 21, 22, 23]. Theorem 2.1. (Bollobás and Erd˝os [4]) For a graph G of size m, √8m +1+1 R(G) , ≥ 4 with equality if and only if G is consists of a complete graph and some isolated vertices. Theorem 2.2. For a connected graph G of order n 2, ψ(G) 2R(G), with ≥ ≤ equality if and only if G ∼= Kn. Proof. Let ψ(G)= k. Then m = e(G) k(k−1) . By Theorem 2.1, ≥ 2 √8m +1+1 p4k(k 1) + 1 + 1 p(2k 1)2 + 1 2k 1 + 1 k R(G) − = − = − = . ≥ 4 ≥ 4 4 4 2 This shows that ψ(G) 2R(G). If k = 2R(G), then m = k(k−1) and the equality ≤ 2 is satisfied. Since G is connected, by Theorem 2.1, G = Kk = Kn. If G = Kn, then ψ(G)= n = 2R(G). Corollary 2.3. For a connected graph G of order n, Γ(G) 2R(G), with equality ≤ if and only if G ∼= Kn. Theorem 2.4. (Wu et al. [28]) If G is a connected graph of order n 2, then ≥ col(G) 2R(G), with equality if and only if G = Kn ≤ ∼ So, combining the above two results, we have Corollary 2.5. For a connected graph G of order n 2, max ψ(G), col(G) ≥ { } ≤ 2R(G), with equality if and only if G ∼= Kn. 3 2.2 Clique number Zaker [30] showed that for a graph G, Γ(G) = 2 if and only if G is a complete bipartite (see also the page 351 in [6]). Zaker and Soltani [34] showed that for any integer k 2, the smallest triangle-free graph of Grundy number k has 2k 2 ≥ − vertices. Let Bk be the graph obtained from Kk−1,k−1 by deleting a matching of cardinality k 2, see Bk for an illustration in Fig. 1. The authors showed that − Γ(Bk)= k. u u u 1 2 k−1 X Y v1v 2 v k−1 Fig. 1. The graph Bk for k 2 ≥ One may formulate the above result of Zaker and Soltani: Γ(G) n+2 for any ≤ 2 triangle-free graph G of order n. Theorem 2.6. (i) For a graph G of order n 1, Γ(G) n+ω(G) . ≥ ≤ 2 (ii) Let k and n be any two integers such that k + n is even and k n. Then there ≤ n+k exists a graph Gk,n on n vertices such that ω(Gk,n)= k and Γ(Gk,n)= 2 . (iii) In particular, if G is a connected triangle-free graph of order n 2, then n+2 ≥ Γ(G)= if and only if G = B n+2 . 2 ∼ 2 Proof. We prove part (i) of the assertion by induction on Γ(G). Let k = Γ(G). If k = 1, then G = Kn. The result is trivially true. Next we assume that k 2. ≥ Let V ,V ,...,Vk be the color classes of a Grundy coloring of G. Set H = G V . 1 2 \ 1 Then Γ(H)= k 1.

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