3 List Coloring Graphs of Girth at least Five on Surfaces

TL;DR

This paper extends key results on 3-list-coloring of graphs with girth at least five to graphs embedded on surfaces, establishing finiteness, linear bounds, and exponential coloring counts, thus broadening the understanding of graph colorability in topological contexts.

Contribution

It generalizes existing planar graph results to graphs on surfaces, proving linear bounds and finiteness of critical graphs, and establishes exponential lower bounds on the number of 3-list-colorings.

Findings

Finiteness of 4-list-critical graphs of girth at least five on fixed surfaces.

Linear bounds on the number of vertices in 4-list-critical graphs related to genus.

Exponential lower bounds on the number of 3-list-colorings for graphs with girth at least five.

Abstract

Grotzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvorak, Kral and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this result to prove Havel's conjecture that a planar graph whose triangles are pairwise far enough apart is 3-colorable. As for list-coloring, Dvorak proved that a planar graph whose cycles of size at most four are pairwise far enough part is 3-choosable. In this article, we generalize these results. First we prove a linear isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. In particular, it follows that there are only finitely many 4-list-critical graphs of girth at least five on any fixed surface, and that in fact the number of vertices of a 4-list-critical graph is linear in its genus. This provides independent proofs of the above results while generalizing Dvorak's result to graphs on surfaces that have large edge-width and yields a similar result showing that a graph of girth at least five with crossings pairwise far apart is 3-choosable. Finally, we generalize to surfaces Thomassen's result that every planar graph of girth at least five has exponentially many distinct 3-list-colorings. Specifically, we show that every graph of girth at least five that has a 3-list-coloring has $2^{\Omega(n)-O(g)}$ distinct 3-list-colorings.