Cycle Double Covers and Semi-Kotzig Frame

TL;DR

This paper introduces semi-Kotzig frames in cubic graphs and proves that the existence of such a frame with limited non-circuit components guarantees a circuit double cover, extending previous results in graph theory.

Contribution

The paper establishes a new sufficient condition for cubic graphs to have a circuit double cover based on semi-Kotzig frames with at most one non-circuit component.

Findings

Cubic graphs with semi-Kotzig frames have circuit double covers.

Generalization of previous results by Goddyn and H"{a}ggkvist and Markstr"{o}m.

Provides a new structural criterion for circuit double covers.

Abstract

Let $H$ be a cubic graph admitting a 3-edge-coloring $c: E(H)\to \mathbb Z_3$ such that the edges colored by 0 and $\mu\in\{1,2\}$ induce a Hamilton circuit of $H$ and the edges colored by 1 and 2 induce a 2-factor $F$. The graph $H$ is semi-Kotzig if switching colors of edges in any even subgraph of $F$ yields a new 3-edge-coloring of $H$ having the same property as $c$. A spanning subgraph $H$ of a cubic graph $G$ is called a {\em semi-Kotzig frame} if the contracted graph $G/H$ is even and every non-circuit component of $H$ is a subdivision of a semi-Kotzig graph. In this paper, we show that a cubic graph $G$ has a circuit double cover if it has a semi-Kotzig frame with at most one non-circuit component. Our result generalizes some results of Goddyn (1988), and H\"{a}ggkvist and Markstr\"{o}m [J. Combin. Theory Ser. B (2006)].