A remark on the Petersen coloring conjecture of Jaeger (Q2848732)

The Petersen coloring conjecture, due to [\textit{F. Jaeger}, in: Selected topics in graph theory, Vol. 3, 71--95 (1988; Zbl 0658.05034)] asserts that if \(G\) is a 2-edge connected cubic graph, then the edges of \(G\) can be colored using the edges of the Petersen graph \(P\) such that any three mutually adjacent edges of \(G\) are colored by three edges that are mutually adjacent in \(P\).NEWLINENEWLINENEWLINE In the present paper a parallel conjecture is posed for the case when the assumption that \(G\) is 2-edge connected is dropped. The new conjecture is named the Sylvester coloring conjecture since the so-called Sylvester graph of order 10 plays the same role as the Petersen graph in the Petersen coloring conjecture. From the main results proved in this paper it follows immediately that the Petersen graph and the Sylvester graph are the only possible graphs in the statements of the conjectures, respectively.