Coloring Complete and Complete Bipartite Graphs from Random Lists

Abstract

Abstract. Assign to each vertex v of the complete graph K n on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f (n)-subsets of a color set [n] = {1, . . . , n}, where f (n) is some integer-valued function of n. Such a list assignment L is called a random (f (n), [n])-list assignment. In this paper, we determine the asymptotic probability (as n → ∞) of the existence of a proper coloring ϕ of K n , such that ϕ(v) ∈ L(v) for every vertex v of K n . We show that this property exhibits a sharp threshold at f (n) = log n. Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph K m,n with parts of size m and n, respectively. We show that if m = o( √ n), f (n) ≥ 2 log n, and L is a random (f (n), [n])-list assignment for the line graph of K m,n , then with probability tending to 1, as n → ∞, there is a proper coloring of the line graph of K m,n with colors from the lists.

On the threshold problem for Latin boxes

“…A similar conjecture was proposed by Casselgren and Häggkvist [6,Conjecture 1.4], although the underlying probability models are different.…”

Coloring graphs of various maximum degree from random lists

“…In , it is proved that for the complete graph K n on n vertices the property of being colorable from a random ‐list assignment has a sharp threshold at at . Moreover, a similar result for the line graph of the complete bipartite graph with parts of size m and n , where is also proved.…”

“…As pointed out above, a rooted k-proper tree has Q(k) vertices, where Q(k) is given by (7). Hence, the number of odd rooted k-proper trees in G is at most n Q(k)−1 .…”

“…Proposition 4.2 Let G = G(n) be a graph on n vertices with maximum degree = (n) and girth at least g, where g > 3 is a fixed positive integer. Suppose further that k is a fixed integer satisfying k ≥ 3 and that L is a random (k, C)-list assignment for G. If σ = ω n 1 Q(k)−1 , where Q(k) is given by (7), then whp G is L-colorable.…”

“…It is straightforward to prove that for odd g, an odd k-proper rooted tree has exactly Q(k) vertices, and that for even g, an even k-proper rooted tree has exactly Q(k) vertices, where Q(k) is given by (7). (Indeed, this formula can be proved by considering a breadth-first search tree with root at some vertex in a graph with minimum degree k.)…”