Sparse Graphs Usually Have Exponentially Many Optimal Colorings

A proper coloring of a graph G = (V,E) is called optimal if the number of colors used is the minimal possible; i.e., it coincides with the chromatic number of G. We investigate the typical behavior of the number of distinct optimal colorings of a random graph G(n, p), for various values of the edge probability p = p(n). Our main result shows that for every constant 1/3 < a < 2, most of the graphs in the probability space G(n, p) with p = n−a have exponentially many optimal colorings. Given a graph G = (V, E), an unordered partition V = V1 ∪ . . . ∪ Vk is called a kcoloring, if each of the color classes Vi is an independent set of G. It is important to observe that we consider unordered partitions only, and therefore two k-colorings (V1, . . . , Vk) and (U1, . . . , Uk), for which there exists a permutation σ ∈ Sk satisfying Vi = Uσ(i), 1 ≤ i ≤ k, are considered to be indistinguishable. A k-coloring (V1, . . . , Vk) of G is optimal, if the number of colors is the minimal possible, i.e. k = χ(G), where χ(G) denotes as usually the chromatic number of G. Here are two simple examples to illustrate the above definitions: a) the graph G = K − e has chromatic number χ(G) = n − 1 and a unique optimal coloring; b) Define G = (V, E) as follows: V = A ∪ B, A ∩ B = ∅, |A| ≥ 1, |B| = n ≥ 2; fix two distinct vertices u, v ∈ B and define E(G) = {(a, b) : a ∈ A, b ∈ B} ∪ {(u, v)}. Then it is easy to see that χ(G) = 3 and G has exactly 2n−2 optimal colorings, where each optimal coloring has the following form: (V1, V2, V3), where V1 = A, u ∈ V2, v ∈ V3. How many optimal colorings does a typical graph G on n vertices with given density p = |E(G)|/ ( n 2 ) have? In order to address this question quantitatively we need to introduce a probability space of graphs on n vertices to make the notion of a “typical graphs” ∗Supported by a USA-Israel BSF grant, by a grant from the Israel Science Foundation and a Bergmann Memorial Award the electronic journal of combinatorics 9 (2002), #R27 1 meaningful. We will make use of the probability space G(n, p) of binomial random graphs. G(n, p) is a random graph on n labeled vertices {1, . . . , n}, where each pair 1 ≤ i < j ≤ n is chosen to be an edge independently and with probability p = p(n). Sometimes with some abuse of notation we will use G(n, p) to denote also a random graph on n vertices chosen according to the distribution induced by G(n, p). As customary we will study asymptotic properties of the random graph G(n, p). This means in particular that the number of vertices n will be assumed as large as necessary. Equipped with this notation we can now reformulate our main question as follows: what is a typical behavior of the number of optimal colorings of a random graph drawn from G(n, p)? As our main result shows this number is exponentially large in n for small and moderate values of the edge probability p = p(n). For simplicity we assume here that p(n) has the form p(n) = n−a for a constant a > 0. Theorem 1 Let > 0. Let p(n) = n−a for a constant a > 0. 1. If 1 3 < a ≤ 1 2 , then with probability at least 1− a random graph G(n, p) has at least exp { 2 10 n 3a−1 2 } optimal colorings; 2. If 1 2 < a < 1, then with probability at least 1− a random graph G(n, p) has at least exp { (1−a) 2 20 n a 2 ln n } optimal colorings. Thus for 1/3 < a < 1 we get exponentially many optimal colorings in a typical graph from G(n, n−a), where the exponent in the estimate of the number of optimal colorings grows with a. To complement the result observe that for all 1 ≤ a < 2 the graph G(n, p) contains almost surely (i.e. with probability tending to 1 as n tends to infinity) Θ(n) isolated vertices and is non-empty. These two conditions imply easily that the number of optimal colorings is e. With some effort Theorem 1 can be strengthened to the “almost sure” form, i.e. the graph G(n, p) will have exponentially many optimal colorings not only with probability at least 1 − , but also almost surely; this would result in some loss in the exponent. Now we will prove our main result, Theorem 1. Denote