Coloring, sparseness, and girth

TL;DR

This paper constructs specialized bipartite graphs with large girth and analyzes their coloring properties, demonstrating sharp bounds for $k$-choosability and providing new constructions for non-colorable graphs with arbitrary girth.

Contribution

It introduces a novel construction of bipartite $r$-augmented trees with large girth and uses these to establish sharp bounds for $k$-choosability in bipartite graphs.

Findings

Maximum average degree at most $2(k-1)$ is sharp for $k$-choosability.

Constructs bipartite graphs with arbitrarily large girth that are non-$k$-choosable.

Provides new simple constructions of non-$k$-colorable graphs and hypergraphs with any girth.

Abstract

An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, 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 $g$.