Proportional 2-Choosability with a Bounded Palette

Abstract

Proportional choosability is a list coloring analogue of equitable coloring. Specifically, a k-assignment L for a graph G specifies a list L(v) of k available colors to each v ∈ V (G). An L-coloring assigns a color to each vertex v from its listMotivated by earlier work, we initiate the study of proportional choosability with a bounded palette by studying proportional 2-choosability with a bounded palette. In particular, when ℓ ≥ 2, a graph G is said to be proportionallyWe show a graph is proportionally (2, 2)-choosable if and only if it is equitably 2-colorable. As ℓ gets larger, the set of propotionally (2, ℓ)-choosable graphs gets smaller. We show that whenever ℓ ≥ 5 a graph is proportionally (2, ℓ)-choosable if and only if it is proportionally 2-choosable. We also completely characterize the connected proportionally (2, ℓ)-choosable graphs when ℓ = 3, 4.

Proportional Choosability of Complete Bipartite Graphs

“…Kaul, Pelsmajer, Reiniger, and the first author [10] introduced a new list analogue of equitable coloring called proportional choosability which places both an upper and lower bound on how many times a color must be used in a list coloring. The study of proportional choosability is in its infancy, but it has received some attention in the literature (see [10,11,19]). Now, following [10] we introduce the specifics.…”