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arXiv · 2406.17723

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

Aseem Dalal·Jessica McDonald·Songling Shan
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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.

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Aseem Dalal, Jessica McDonald, Songling Shan. 2024-06-25. A reduction of the "cycles plus $K_4$'s" problem. https://arxiv.org/abs/2406.17723

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