Incidence Posets and Cover Graphs

TL;DR

This paper explores the relationships between incidence posets, cover graphs, and graph parameters, revealing that chromatic number is unbounded by incidence poset dimension and identifying graphs with large girth and chromatic number within certain classes.

Contribution

It proves that graph chromatic number is not bounded by the dimension of its incidence poset for dimensions four and above, and identifies graphs with large girth and chromatic number with low eye parameter.

Findings

Chromatic number unbounded by incidence poset dimension for dimension ≥ 4

Existence of graphs with large girth and chromatic number with low eye parameter

Answers to questions posed by Haxell and Kříž & Nešetřil

Abstract

We prove two theorems concerning incidence posets of graphs, cover graphs of posets and a related graph parameter. First, answering a question of Haxell, we show that the chromatic number of a graph is not bounded in terms of the dimension of its incidence poset, provided the dimension is at least four. Second, answering a question of K\v{r}\'{i}\v{z} and Ne\v{s}et\v{r}il, we show that there are graphs with large girth and large chromatic number among the class of graphs having eye parameter at most two.