Cycle Double Covers and Semi-Kotzig Frame

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)]

Perfect Matching and Circuit Cover of Graphs

The research of my dissertation is motivated by the Circuit Double Cover Conjecture due to Szekeres and independently Seymour, that every bridgeless graph G has a family of circuits which covers every edge of G twice. By Fleischner\u27s Splitting Lemma, it suffices to verify the circuit double cover conjecture for bridgeless cubic graphs.;It is well known that every edge-3-colorable cubic graph has a circuit double cover. The structures of edge-3-colorable cubic graphs have strong connections with the circuit double cover conjecture. In chapter two, we consider the structure properties of a special class of edge-3-colorable cubic graphs, which has an edge contained by a unique perfect matching. In chapter three, we prove that if a cubic graph G containing a subdivision of a special class of edge-3-colorable cubic graphs, semi-Kotzig graphs, then G has a circuit double cover.;Circuit extension is an approach posted by Seymour to attack the circuit double cover conjecture. But Fleischer and Kochol found counterexamples to this approach. In chapter four, we post a modified approach, called circuit extension sequence. If a cubic graph G has a circuit extension sequence, then G has a circuit double cover. We verify that all Fleischner\u27s examples and Kochol\u27s examples have a circuit extension sequence, and hence not counterexamples to our approach. Further, we prove that a circuit C of a bridgeless cubic G is extendable if the attachments of all odd Tutte-bridges appear on C consequently.;In the last chapter, we consider the properties of minimum counterexamples to the strong circuit double cover. Applying these properties, we show that if a cubic graph G has a long circuit with at least | V(G)| - 7 vertices, then G has a circuit double cover