4-critical graphs on surfaces without contractible (<=4)-cycles

TL;DR

This paper characterizes 4-critical graphs embedded on surfaces with no small contractible cycles, showing they can be decomposed into a bounded part and unbounded components, with the proof aided by computer enumeration.

Contribution

It provides a structural decomposition of 4-critical graphs on surfaces without small contractible cycles, using computer-assisted enumeration for the base case.

Findings

Decomposition of 4-critical graphs into bounded and unbounded parts

Bound on the size of the bounded part depends linearly on surface genus

Computer enumeration of plane 4-critical graphs with girth 5 and small precolored cycles

Abstract

We show that if G is a 4-critical graph embedded in a fixed surface $\Sigma$ so that every contractible cycle has length at least 5, then G can be expressed as $G=G'\cup G_1\cup G_2\cup ... \cup G_k$, where $|V(G')|$ and $k$ are bounded by a constant (depending linearly on the genus of $\Sigma$) and $G_1\ldots,G_k$ are graphs (of unbounded size) whose structure we describe exactly. The proof is computer-assisted - we use computer to enumerate all plane 4-critical graphs of girth 5 with a precolored cycle of length at most 16, that are used in the basic case of the inductive proof of the statement.