Coloring graphs from random lists of size 2

Let G = G ( n ) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Δ . Assign to each vertex v of G a list L ( v ) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size σ ( n ) . In this paper we determine, for each fixed g and growing n , the asymptotic probability of the existence of a proper coloring φ such that φ ( v ) ∈ L ( v ) for all v ∈ V ( G ) . In particular, we show that if g is odd and σ ( n ) = ω ( n 1 / ( 2 g − 2 ) ) , then the probability that G has a proper coloring from such a random list assignment tends to 1 as n → ∞ . Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n ≥ g , there is a graph H = H ( n , g ) with bounded maximum degree and girth g , such that if σ ( n ) = o ( n 1 / ( 2 g − 2 ) ) , then the probability that H has a proper coloring from such a random list assignment tends to 0 as n → ∞ . A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size σ ( n ) , exhibits a sharp threshold at σ ( n ) = 2 n .