A note on the shameful conjecture

Abstract

Abstract. Let P G (q) denote the chromatic polynomial of a graph G on n vertices. The 'shameful conjecture' due to Bartels and Welsh states that,Let µ(G) denote the expected number of colors used in a uniformly random proper n-coloring of G. The above inequality can be interpreted as saying that µ(G) ≥ µ(On), where On is the empty graph on n nodes. This conjecture was proved by F. M. Dong, who in fact showed that,There are examples showing that this inequality is not true for all q ≥ 2. In this paper, we show that the above inequality holds for all q ≥ 36D 3/2 , where D is the largest degree of G. It is also shown that the above inequality holds true for all q ≥ 2 when G is a claw-free graph.

Collision times in multicolor urn models and sequential graph coloring with applications to discrete logarithms

“…, p c ), where the probability that a vertex is colored with color a ∈ [c] is p a which is independent of the colors of the other vertices, where p a ≥ 0, and c a=1 p a = 1. Then the probability that G is properly colored is related to Stanley's generalized chromatic polynomial [19,43]. Limit theorems for the number of monochromatic edges under the uniform coloring distribution, that is, p a = 1/c for all a ∈ [c], was derived recently by Bhattacharya et al [8].…”