Even Subdivision-Factors of Cubic Graphs

TL;DR

This paper investigates the properties of even subdivision-factors in cubic graphs, disproves a related conjecture, and contributes to the understanding of the circuit double cover conjecture.

Contribution

It demonstrates that certain sets of 2-connected graphs cannot serve as universal even subdivision-factors for all 3-connected cubic graphs, disproving a prior conjecture.

Findings

Disproved a conjecture related to even subdivision-factors.

Identified properties of sets that are even subdivision-factors.

Contributed to the understanding of the circuit double cover conjecture.

Abstract

We call a set $\mathcal S$ of graphs an "even subdivison-factor" of a cubic graph $G$ if $G$ contains a spanning subgraph $H$ such that every component of $H$ has an even number of vertices and is a subdivision of an element of $\mathcal S$. We show that any set of 2-connected graphs which is an even subdivison-factor of every 3-connected cubic graph, satisfies certain properties. As a consequence, we disprove a conjecture which was stated in an attempt to solve the circuit double cover conjecture.