Local chromatic number of quadrangulations of surfaces

TL;DR

This paper explores how the local chromatic number of odd quadrangulations varies with the genus of non-orientable surfaces, revealing genus-dependent behavior unlike the classical chromatic number.

Contribution

It demonstrates that the local chromatic number's lower bounds depend on the surface's genus, contrasting with the genus-independent bounds for the chromatic number.

Findings

For non-orientable surfaces of genus ≤4, odd quadrangulations have local chromatic number at least four.

For genus ≥5, there exist odd quadrangulations with local chromatic number less than four.

Face subdivisions and Fisk triangulations behave similarly for local and usual chromatic numbers.

Abstract

The local chromatic number of a graph was introduced by Erd\H{o}s et al. [4]. In [17] a connection to topological properties of (a box complex of) the graph was established and in [18] it was shown that if a graph is strongly topologically 4-chromatic then its local chromatic number is at least four. As a consequence one obtains a generalization of the following theorem of Youngs: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs's result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number.