Normal edge-colorings of cubic graphs

A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors having the additional property that when looking at the set of colors assigned to any edge $e$ and the four edges adjacent it, we have either exactly five distinct colors or exactly three distinct colors. We denote by $\chi'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal $k$-edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. More precisely, it is known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture and then, among others, Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the larger class of all simple cubic graphs (not necessarily bridgeless), some interesting questions naturally arise. For instance, there exist simple cubic graphs, not bridgeless, with $\chi'_{N}(G)=7$. On the other hand, the known best general upper bound for $\chi'_{N}(G)$ was $9$. Here, we improve it by proving that $\chi'_{N}(G)\leq7$ for any simple cubic graph $G$, which is best possible. We obtain this result by proving the existence of specific no-where zero $\mathbb{Z}_2^2$-flows in $4$-edge-connected graphs.Comment: 17 pages, 6 figure

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

Normal 5-edge coloring of some more snarks superpositioned by the Petersen graph

A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement that every bridgeless cubic graph has a normal 5-edge-coloring. All 3-edge-colorings of a cubic graph are obviously normal, so in order to establish the conjecture it is sufficient to consider only snarks. In our previous paper [J. Sedlar, R. \v{S}krekovski, Normal 5-edge-coloring of some snarks superpositioned by the Petersen graph, Applied Mathematics and Computation 467 (2024) 128493], we considered superpositions of any snark G along a cycle C by two simple supervertices and by the superedge obtained from the Petersen graph, but only for some of the possible ways of connecting supervertices and superedges. The present paper is a continuation of that paper, herein we consider superpositions by the Petersen graph for all the remaining connections and establish that for all of them the Petersen Coloring Conjecture holds.Comment: 19 pages, 10 figure

Normal 5-edge-coloring of some snarks superpositioned by Flower snarks

An edge e is normal in a proper edge-coloring of a cubic graph G if the number of distinct colors on four edges incident to e is 2 or 4: A normal edge-coloring of G is a proper edge-coloring in which every edge of G is normal. The Petersen Coloring Conjecture is equivalent to stating that every bridgeless cubic graph has a normal 5-edge-coloring. Since every 3-edge-coloring of a cubic graph is trivially normal, it is suficient to consider only snarks to establish the conjecture. In this paper, we consider a class of superpositioned snarks obtained by choosing a cycle C in a snark G and superpositioning vertices of C by one of two simple supervertices and edges of C by superedges Hx;y, where H is any snark and x; y any pair of nonadjacent vertices of H: For such superpositioned snarks, two suficient conditions are given for the existence of a normal 5-edge-coloring. The first condition yields a normal 5-edge-coloring for all hypohamiltonian snarks used as superedges, but only for some of the possible ways of connecting them. In particular, since the Flower snarks are hypohamiltonian, this consequently yields a normal 5-edge-coloring for many snarks superpositioned by the Flower snarks. The second sufficient condition is more demanding, but its application yields a normal 5-edge-colorings for all superpositions by the Flower snarks. The same class of snarks is considered in [S. Liu, R.-X. Hao, C.-Q. Zhang, Berge{Fulkerson coloring for some families of superposition snarks, Eur. J. Comb. 96 (2021) 103344] for the Berge-Fulkerson conjecture. Since we established that this class has a Petersen coloring, this immediately yields the result of the above mentioned paper.Comment: 30 pages, 16 figure