Coloring near-quadrangulations of the cylinder and the torus

Abstract

Let G be a simple connected plane graph and let C1 and C2 be cycles in G bounding distinct faces f1 and f2. 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 rf1,f2(G)=∏fr(|f|), where the product is over the faces f of G distinct from f1 and f2, and let q(G)=1+∑f:|f|≠4|f|, where the sum is over all faces f of G (of length other than four). We give an algorithm with time complexity O(rf1,f2(G)q(G)|G|) which, given a 3-coloring ψ of C1∪C2, either finds an extension of ψ 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.