Coloring, sparseness and girth

Abstract

An r-augmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d, g, r ∈ N, we construct a bipartite r-augmented complete d-ary tree having girth at least g. The height of such trees must grow extremely rapidly in terms of the girth.Using the resulting graphs, we construct sparse non-k-choosable bipartite graphs, showing that maximum average degree at most 2(k − 1) is a sharp sufficient condition for k-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-k-colorable graphs and hypergraphs with any girth g.

List Coloring with a Bounded Palette

“…To start with the smallest open case, is there an * such that every (3, * )-choosable graph is 4-choosable? If so, is the value suggested by Table I optimal, i.e., is it true that every (3,9)-choosable graph is 4-choosable?…”

“…Recently, Alon et al. sharpened the boundary between and by exhibiting high girth, bipartite, non‐‐choosable graphs all proper subgraphs of which have average degree at most .…”

“…Observe that the condition ≥ 2k − 1 cannot be ignored, because every bipartite graph is (k, 2k − 2)-choosable and the class of bipartite graphs has unbounded choosability. Recently, Alon et al [3] sharpened the boundary between = 2k − 2 and = 2k − 1 by exhibiting high girth, bipartite, non-(k, 2k − 1)-choosable graphs all proper subgraphs of which have average degree at most 2k − 2.…”

Fractional DP‐colorings of sparse graphs

“…For completeness, we sketch it in Section 5. Another related family of graphs of high chromatic number that falls under the conditions of Theorem 1.10. is described in [1,Theorem 3.4]. From these constructions, we deduce the following result:  , there exists a graph G d g , with chromatic number at least d, girth at least g, and…”

“…Another related family of graphs of high chromatic number that falls under the conditions of Theorem . is described in , Theorem 3.4]. From these constructions, we deduce the following result:…”