Some new results on Sylvester colorings of cubic graphs

If $G$ and $H$ are two cubic multi-graphs, then an $H$-coloring of $G$ is a mapping $f: E(G)\rightarrow E(H)$, such that for every $v\in V(G)$ there is a vertex $x\in V(H)$, such that $f(\partial_G(v))=\partial_H(x)$. If $G$ admits an $H$-coloring then it is common to write $H\prec G$. The Petersen coloring conjecture predicts that for any bridgeless cubic graph $G$ one has $P_{10}\prec G$. Here $P_{10}$ is the Petersen graph. Let $f: E(G)\rightarrow E(H)$ be any mapping. Define: $V(f)=\{v\in V(G):\exists x\in V(H), f(\partial_G(v))=\partial_H(x)\}$. Let $S_{10}$ be the smallest cubic multi-graph that has no perfect matching. It has ten vertices. Define $S_{12}$ as the cubic graph that is obtained from $S_{10}$, by replacing its unique vertex $z$ adjacent to three bridges with a triangle. In this paper we show that (1) for every cubic multi-graph $G$ with a perfect matching, there is a mapping $f:E(G)\rightarrow E(S_{12})$, such that $|V(f)|\geq \frac{4}{5}\cdot |V(G)|$, and (2) for every cubic multi-graph $G$, there is a mapping $f:E(G)\rightarrow E(S_{10})$, such that $|V(f)|\geq \frac{5}{6}\cdot |V(G)|$. Our second result improves the $\frac{4}{5}$-bound by Hakobyan and the second author from 2018.