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List-Distinguishing Cartesian Products of Cliques San Jose State University From the SelectedWorks of Sogol Jahanbekam July, 2019 List-Distinguishing Cartesian Products of Cliques Michael Ferrara, University of Colorado, Denver Zoltan Füredi, Alfréd Rényi Institute of Mathematics Sogol Jahanbekam, Rochester Institute of Technology Paul Wenger, Rochester Institute of Technology Available at: https://works.bepress.com/sogol-jahanbekam/2/ List-Distinguishing Cartesian Products of Cliques Michael Ferrara∗, Zoltan F¨uredi†, Sogol Jahanbekam‡, and Paul S. Wenger‡ October 16, 2018 Abstract The distinguishing number of a graph G, denoted D(G), is the minimum number of colors needed to produce a coloring of the vertices of G so that every nontrivial isomorphism interchanges vertices of different colors. A list assignment L on a graph G is a function that assigns each vertex of G a set of colors. An L-coloring of G is a coloring in which each vertex is colored with a color from L(v). The list distinguishing number of G, denoted Dℓ(G) is the minimum k such that every list assignment L that assigns a list of size at least k to every vertex permits a distinguishing L-coloring. In this paper, we prove that when and n is large enough, the distinguishing and list- distinguishing numbers of Kn✷Km agree for almost all m>n, and otherwise differ by at most one. As a part of our proof, we give (to our knowledge) the first application of the Combinatorial Nullstellensatz to the graph distinguishing problem and also prove an inequality for the binomial distribution that may be of independent interest. Keywords: distinguishing, list distinguishing, 05C60, 05C15 1 Introduction Given a graph G, a k-coloring φ : X 1,...,k of G is distinguishing if the only auto- → { } morphism of G that fixes φ is the identity automorphism. The minimum k for which G has arXiv:1707.01823v1 [math.CO] 6 Jul 2017 a distinguishing k-coloring is called the distinguishing number of G, and is denoted D(G). The distinguishing number of a graph G is, in some sense, a measure of the resilience of Aut(G), in that a distinguishing coloring serves to “break” all of the symmetries of G. ∗Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO; [email protected]. Research was supported in part by Simons Foundation Collaboration Grants #206692 and #426971. †Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary; [email protected]. Research was supported in part by grant K116769 from the National Research, Development and Innovation Office NKFIH, and by the Simons Foundation Collaboration Grant #317487. ‡School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY; [email protected], [email protected]. 1 Inspired by a problem of Rubin [22], which creatively asked for the distinguishing number of the cycle Cn, Albertson and Collins initiated the study of distinguishing numbers in [2]. Since that initial work, the distinguishing number of a graph has been determined for a number of graph classes, including planar graphs [4], Cartesian powers [1, 20], forests [11] and interval graphs [10]. The distinguishing problem has also been studied extensively for infinite graphs (see [21] for a recent example). In 2011, Ferrara, Flesch, and Gethner [13] first considered the natural extension of graph distinguishing to list colorings. A list assignment L on a graph G is a function that assigns each vertex of G a set of colors. An L-coloring of G is a coloring in which each vertex is colored with a color from L(v). List coloring was first introduced in the setting of proper colorings by Erd˝os, Rubin, and Taylor [12], and has been studied extensively across numerous settings. The list-distinguishing number of G, denoted Dℓ(G) is the minimum k such that every list assignment L that assigns a list of size at least k to every vertex permits a distinguishing L-coloring. In [13], Ferrara, Flesch, and Gethner posed the following question. Question 1. Is there a graph G for which D(G) = D (G)? 6 ℓ Question 1 is unanswered, but in subsequent years, there has been an accumulation of evidence suggesting that the negative answer to Question 1 is correct. This includes proofs that D(G) = Dℓ(G) when G has a dihedral automorphism group [13], is a forest [14], and when G is an interval graph [17]. We note that the problem of how one can distinguish the vertices of a graph, be it through coloring or the identification of a “special” set of vertices, has been broadly studied. As a significant recent example, Babai’s proof of the existence of a near polynomial-time algorithm for the graph isomorphism problem [6] relies on the idea of a distinguishing set of vertices, which is a set S of vertices such that every vertex not in the set has a unique subset of S in its neighborhood [7]. Further examples include identifying codes [19], many variants of vertex-distinguishing edge-colorings [9], and a number of other concepts throughout the literature. 2 Results In this paper, we study the list-distinguishing number of Cartesian products of complete graphs. The distinguishing number for Cartesian products of complete graphs was deter- mined independently by Imrich, Jerebic, and Klavˇzar [18] and also Fisher and Isaak [16]. 2 Interestingly, Fisher and Isaak’s result was not phrased in terms of graph distinguishing, but rather in the setting of a particular class of edge colorings of the complete bipartite graph. Theorem 1 ([18]). Let n, m, k be positive integers with k 2. If (k 1)n < m kn, then ≥ − ≤ n k if m k logk n 1; D(Kn✷Km)= ≤ − ⌈ ⌉ − k +1 if m kn log n +1. ( ≥ − ⌈ k ⌉ If m = kn log n , then D(K ✷K ) is either k or k +1 and can be computed recursively − ⌈ k ⌉ n m in O(log∗(m)) time. Our main result is the following. Theorem 2. Let n, m, and k be positive integers. If n < m kn(1 log1.09 n ), then ≤ − √2n D (K ✷K ) k. ℓ n m ≤ Comparing Theorems 1 and 2 in light of Question 1, we see that Theorem 2 shows that for n sufficiently large, the distinguishing and list-distinguishing numbers of Kn✷Km agree for almost all m, and otherwise differ by at most one. Throughout the paper we treat the vertices of Kn✷Km as the points on an integer lattice with n rows and m columns. Each copy of Km in the product will correspond to a row of the lattice, and each copy of Kn in the product will correspond to a column of the lattice (see Figure 1). The vertices will be labeled by v with 1 i n and 1 j m, with i,j ≤ ≤ ≤ ≤ vertex v1,1 in the top left corner and vertex vn,m in the bottom right corner (like the entries of an n m matrix). × To prove Theorem 2, we begin by determining the list distinguishing number of Kn✷Kn+1. We prove this using the Combinatorial Nullstellensatz. Theorem 3 (Alon [3]). Let F be an arbitrary field, and let f = f(x1,...,xn) be a polynomial n in F [x1,...,xn]. Suppose that the degree of f is i=1 ti where each ti is a nonnegative integer, and suppose that the coefficient of n xti in f is nonzero. If S ,...,S are subsets of F i=1 i P 1 n with S > t , then there are s S ,s S ,...,s S so that | i| i 1 ∈ Q1 2 ∈ 2 n ∈ n f(s ,...,s ) =0. 1 n 6 n ti n ti We adopt the notation f i=1 xi to denote the coefficient of i=1 xi in f. Lemma 4. For n 1, D (KQ✷K )=2 . Q ≥ ℓ n n+1 3 Km ... v1,1 v1,2 v1,m Km ... v2,1 v2,2 Km ... .. Km ... vn,1 vn,m Kn Kn Kn Kn Kn Figure 1: Kn✷Km. Proof. By Theorem 1, we know that D(K ✷K ) = 2. Since D (G) D(G) for all graphs, n n+1 ℓ ≥ it follows that D (K ✷K ) 2. Let L be a list assignment to V (K ✷K ) in which ℓ n n+1 ≥ n n+1 every list has two distinct elements, and assume that those elements are from R. It remains to show that there is a distinguishing L-coloring of Kn✷Kn+1. To apply the Combinatorial Nullstellensatz, we will create a polynomial F of degree n2 in n(n+1) variables such that nonzero valuations of F correspond to distinguishing L-colorings of K ✷K . The variables of the polynomial are x i [n], j [n + 1] , and the value n n+1 { i,j | ∈ ∈ } of xi,j will be taken from L(vi,j). To build the polynomial F , we first define two families of polynomials whose product is F . The first family is used to differentiate the columns of the graph. The second family will then be used to differentiate the rows of the graph. For 1 i < j n + 1, define ≤ ≤ n n C = (x ) (x ), i,j k,j − k,i Xk=1 Xk=1 and set C = 1 i<j n+1 Ci,j. For 1 h<l n, define ≤ ≤ ≤ ≤ Q R = x x h,l l,h − h,h and set R = 1 h<l n Rh,l. Finally, define ≤ ≤ Q F = C R. · 4 We claim that a nonzero valuation of F corresponds to a distinguishing coloring of K ✷K . In such a valuation, C = 0 for all 1 i < j n + 1, and R = 0 for n n+1 i,j 6 ≤ ≤ h,l 6 1 h<l n.
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