The maximum number of colorings of graphs of given order and size

The set of all simple graphs on n vertices and m edges f (n, m, λ) The maximum number of coloring of a graph with n vertices and m edges in at most λ colors (n, m)-graph A graph of order n and size m i 1 The maximum number of colorings of graphs of given order and size Below is the plan of this chapter. 1 2 The maximum number of colorings of graphs of given order and size • Definitions and notation • Background and motivations • Results • Ideas and techniques • Some open questions 1.2 Introduction Consider a family F n,m of all simple graphs on n vertices and m edges. Let λ be a positive integer. What is the maximum number of proper vertex colorings in (at most) λ colors a graph from F n,m can have? On which graphs is this maximum attained? Clearly, the question is equivalent of finding f (n, m, λ) = max{χ(G, λ) : G ∈ F n,m }, where χ(G, λ) is the chromatic polynomial of G. In probabilistic terms, the question is equivalent to the following. Consider a graph G ∈ F n,m , and color its vertices uniformly at random in a given set of λ colors. What is the maximum probability of obtaining a proper coloring of G and what are the extremal graphs? The question was asked independently by H.S. Wilf [40] and N. Linial [16], and is still unsolved. All graphs in this paper are finite, undirected, and have neither loops nor multiple edges. For all missing definitions and basic facts which are mentioned but not proved, we refer the reader to Bollobás [5]. For a graph G, let V = V (G) and E = E(G) denote the vertex set of G and the edge set of G, respectively. Let |A| denote the cardinality of a set A. Let n = n(G) = |V (G)| and m = m(G) = |E(G)| denote the number of vertices (the order) of G, and number of edges (the size) of G, respectively. Let F n,m denote the set of all graphs with n vertices and m edges. An edge {x, y} of G will also be denoted by xy, or yx. For sets X, Y , let X − Y = X \ Y. For A ⊆ V (G), let G[A] denote the subgraph of G …