The Adaptable Chromatic Number and the Chromatic Number

Abstract

We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible.

Single-conflict colorings of degenerate graphs

“…), containing in particular those d-degenerate graphs G with maximum degree ∆ = exp(d o (1) ). This upper bound is close to best possible, since Molloy's lower bound from [8] shows that…”

“…√ ∆ [10]. Molloy shows furthermore that graphs G with chromatic number χ(G) satisfy χ ad (G) ≥ (1 + o(1)) χ(G) [8], implying that χ(G), χ ad (G), χ (G), and √ ∆ all only differ by a constant factor for graphs G satisfying χ(G) = Θ(∆). The parameters χ(G), χ ad (G), and χ (G) can also differ by a constant factor even when χ(G) is not of the form Θ(∆).…”

Adaptable and conflict colouring multigraphs with no cycles of length three or four

“…by (2). Therefore, by the law of total probability and ( 13), ( 21), (22), we get that the probability of the event A occurring is:…”

“…For any g and ∆ ≥ 3, random ∆regular graphs have girth at least g and satisfy χ(G) ≥ ( 1 2 + o(1)) ∆ ln ∆ with high probability [14]. Moreover it is known that for any graph G, χ a (G) ≥ (1+o(1)) χ(G) by [22]. Combining these two facts implies the existence of a graph G with maximum degree ∆ and no 3 or 4-cycles, satisfying χ a (G) ≥ ( √ 2 2 + o(1)) ∆/ ln ∆.…”

Cooperative colorings of forests

“…A cooperative coloring problem on a family G may be translated into an adapted coloring problem by coloring the edges of each graph G i ∈ G with the color i and then considering the multigraph obtained from the union of all graphs in G, and an adapted coloring problem may be similarly translated into a cooperative coloring problem. Adapted colorings were first considered by Kostochka and Zhu [9] and have been frequently studied since then [6,8,11,13].…”