A note on choosability with defect 1 of graphs on surfaces

TL;DR

This paper establishes a new upper bound on the choosability with defect 1 for graphs on surfaces, showing that allowing a small defect reduces the number of colors needed compared to traditional coloring.

Contribution

It proves that all graphs on a surface of Euler genus μ are eil 2 + \u221a{3} -choosable with defect 1, improving understanding of surface graph colorability with minimal defect.

Findings

Surface graphs are eil 2 + {3} -choosable with defect 1.

Toroidal graphs are 5-choosable with defect 1.

This reduces the chromatic number requirement for surface graphs with minimal defect.

Abstract

This note proves that every graph of Euler genus $\mu$ is $\lceil 2 + \sqrt{3\mu + 3}\,\rceil$--choosable with defect 1 (that is, clustering 2). Thus, allowing defect as small as 1 reduces the choice number of surface embeddable graphs below the chromatic number of the surface. For example, the chromatic number of the family of toroidal graphs is known to be $7$. The bound above implies that toroidal graphs are $5$-choosable with defect 1. This strengthens the result of Cowen, Goddard and Jesurum (1997) who showed that toroidal graphs are $5$-colourable with defect 1.