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If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a properedge-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 coloringconjecture of Jaeger ($P_{10}$-conjecture) states that for any bridgeless cubicgraph $G$, one has: $P_{10}\prec G$. The Sylvester coloring conjecture($S_{10}$-conjecture) states that for any cubic graph $G$, $S_{10}\prec G$. Inthis paper, we introduce two new conjectures that are related to theseconjectures. The first of them states that any cubic graph with a perfectmatching admits an $S_{12}$-coloring. The second one states that any cubicgraph $G$ whose edge-set can be covered with four perfect matchings, admits a$P_{12}$-coloring. We call these new conjectures $S_{12}$-conjecture and$P_{12}$-conjecture, respectively. Our first results justify the choice ofgraphs in $S_{12}$-conjecture and $P_{12}$-conjecture. Next, we characterizethe edges of $P_{12}$ that may be fictive in a $P_{12}$-coloring of a cubicgraph $G$. Finally, we relate the new conjectures to the already knownconjectures by proving that $S_{12}$-conjecture implies $S_{10}$-conjecture,and $P_{12}$-conjecture and $(5,2)$-Cycle cover conjecture together imply$P_{10}$-conjecture. Our main tool for proving the latter statement is a newreformulation of $(5,2)$-Cycle cover conjecture, which states that the edge-setof any claw-free bridgeless cubic graph can be covered with four perfectmatchings.

$S_{12}$ and $P_{12}$-colorings of cubic graphs