On coupon colorings of graphs

Let G be a graph with no isolated vertices. A k-coupon coloring of G is an assignment of colors from [k] := {1, 2, . . . , k} to the vertices of G such that the neighborhood of every vertex of G contains vertices of all colors from [k]. The maximum k for which a k-coupon coloring exists is called the coupon coloring number of G, and is denoted χc(G). In this paper, we prove that every d-regular graph G has χc(G) ≥ (1 − o(1))d/ log d as d → ∞, and the proportion of d-regular graphs G for which χc(G) ≤ (1 + o(1))d/ log d tends to 1 as |V (G)| → ∞.