A reduction of the "cycles plus $K_4$'s" problem

TL;DR

This paper reduces the complex 'cycles plus K4s' problem to a more manageable 3-colourability problem involving specific graph constructions, advancing understanding of graph colourability related to the Strong Colouring Conjecture.

Contribution

It introduces a reduction from the 'cycles plus K4s' problem to a specialized 3-colourability problem, simplifying the original problem.

Findings

Reduction from 'cycles plus K4s' to 3-colourability problem

Defines specific graph constructions involving triangles and paths

Provides a new approach to the Strong Colouring Conjecture

Abstract

Let $H$ be a 2-regular graph and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. The "cycles plus $K_4$'s" problem is to show that $G$ is 4-colourable; this is a special case of the \emph{Strong Colouring Conjecture}. In this paper we reduce the "cycles plus $K_4$'s" problem to a specific 3-colourability problem. In the 3-colourability problem, vertex-disjoint triangles are glued (in a limited way) onto a disjoint union of triangles and paths of length at most 12, and we ask for 3-colourability of the resulting graph.