A remark on Petersen coloring conjecture of Jaeger

If $G$ and $H$ are two cubic graphs, then we write $H\prec G$, if $G$ admits a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. Let $P$ and $S$ be the Petersen graph and the Sylvester graph, respectively. In this paper, we introduce the Sylvester coloring conjecture. Moreover, we show that if $G$ is a connected bridgeless cubic graph with $G\prec P$, then $G=P$. Finally, if $G$ is a connected cubic graph with $G\prec S$, then $G=S$.Comment: 6 pages, 2 figures, + the comments of the referees are taken into account+ Sylvester coloring conjecture is introduced+ a result related with this conjectures is prove

On Sylvester Colorings of Cubic Graphs

If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger states that for any bridgeless cubic graph $G$, one has: $P\prec G$. The second author has recently introduced the Sylvester coloring conjecture, which states that for any cubic graph $G$ one has: $S\prec G$. Here $S$ is the Sylvester graph on $10$ vertices. In this paper, we prove the analogue of Sylvester coloring conjecture for cubic pseudo-graphs. Moreover, we show that if $G$ is any connected simple cubic graph $G$ with $G\prec P$, then $G = P$. This implies that the Petersen graph does not admit an $S_{16}$-coloring, where $S_{16}$ is the smallest connected simple cubic graph without a perfect matching. $S_{16}$ has $16$ vertices. %We conjecture that there are infinitely many connected cubic simple graphs which do not admit an %$S_{16}$-coloring. Finally, we obtain $2$ results towards the Sylvester coloring conjecture. The first result states that any cubic graph $G$ has a coloring with edges of Sylvester graph $S$ such that at least $\frac45$ of vertices of $G$ meet the conditions of Sylvester coloring conjecture. The second result states that any claw-free cubic graph graph admits an $S$-coloring. This results is an application of our result on cubic pseudo-graphs.Comment: 18 pages, 14 figure