A counterexample to the S10- and the S12-Conjecture

For two graphs G and H, a mapping f: E(G) → E(H) is an H-coloring of G, if it is a proper edge-coloring and for every v ∈ V(G) there exists a vertex u ∈ V(H) with f(∂_(G)(v)) = ∂_(H)(u). Motivated by the Petersen Coloring Conjecture, paper from Mkrtchyan (2013) and paper from Mkrtchyan together with Hakobyan (2019); made the following two conjectures. (I) Every cubic graph has an S₁₀-coloring, where S₁₀ is a graph on 10 vertices sometimes also referred to as the Sylvester graph. (II) Every cubic graph with a perfect matching has an S₁₂-coloring, where S₁₂ is the graph obtained from S₁₀ by replacing the central vertex with a triangle. In this note we present a (rather small) counterexample to both conjectures.