Coloring near-quadrangulations of the cylinder and the torus

TL;DR

This paper presents an algorithm for extending 3-colorings in near-quadrangulations of the cylinder and torus, based on a min-max theorem for integer flows, with implications for graph colorability.

Contribution

It introduces a novel algorithm with complexity depending on face structure and provides a new obstruction criterion for 3-colorability in toroidal graphs.

Findings

Algorithm determines extendability of 3-colorings in near-quadrangulations.

Establishes a min-max theorem for a variant of integer 2-commodity flows.

Shows that triangle-free graphs on the torus with edge-width ≥ 21 are 3-colorable.

Abstract

Let G be a simple connected plane graph and let C_1 and C_2 be cycles in G bounding distinct faces f_1 and f_2. For a positive integer l, let r(l) denote the number of integers n such that -l<=n<=l, n is divisible by 3, and n has the same parity as l; in particular, r(4)=1. Let r_{f_1,f_2}(G) be the product of r(|f|) over all faces f of G distinct from f_1 and f_2, and let q(G)=1+sum_{f:|f|\neq 4} |f|, where the sum is over all faces f of G. We give an algorithm with time complexity O(r_{f_1,f_2}(G)q(G)|G|) which, given a 3-coloring psi of C_1 and C_2, either finds an extension of psi to a 3-coloring of G, or correctly decides no such extension exists. The algorithm is based on a min-max theorem for a variant of integer 2- commodity flows, and consequently in the negative case produces an obstruction to the existence of the extension. As a corollary, we show that every triangle-free graph drawn in the torus with edge-width at least 21 is 3-colorable.