Normal 6-edge-colorings of some bridgeless cubic graphs

In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors such that each edge of the graph is normal. We denote by $\chi'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal $7$-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal $6$-edge-coloring. Finally, we show that any bridgeless cubic graph $G$ admits a $6$-edge-coloring such that at least $\frac{7}{9}\cdot |E|$ edges of $G$ are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with arXiv:1804.0944

Graph coloring and flows

Part 1. The Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, we verify the conjecture for some families of snarks (Goldberg snarks, flower snarks) by using a technical lemma.;Part 2. A star coloring of an undirected graph G is a proper vertex coloring of G such that any path of length 3 in G is not bi-colored. The star chromatic number of a family of graphs G , denoted by chis( G ), is the minimum number of colors that are necessary to star color any graph belonging to G . Let FD be the family of all graphs with maximum degree at most Delta. It was proved by G. Fertin, A. Raspaud and B. Reed (JGT 2004) that chi s( FD ) ≥ 2Delta where 1 ≤ Delta ≤ 3. In this paper, this result is further generalized for every positive integer Delta. That is, chi s FD ≥ 2Delta for every Delta ∈ Z+ . It was proved by M. Albertson, G. Chappell, H. Kierstead, A. Kundgen, R. Ramamurthi (EJC 2004) that chis( FD ) ≤ Delta(Delta - 1) + 2. In this paper, a simplified proof is given and this result is further improved for non Delta-regular graph to chis FngD ≤ Delta(Delta - 1) + 1 where FngD is the family of non-regular graphs with maximum degree Delta.;Part 3. There are two famous conjectures about integer flows, the 5-flow conjecture raised by Tutte and the orientable 5-cycle double cover by Archdeacon [1] and Jaeger [15]. It is known that the orientable 5-cycle double cover conjecture implies the 5-flow conjecture. But the converse is not known to hold. In this paper, we try to use the reductions of incomplete integer flows to lead us in a direction to attack the problem

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