^{}Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui, 233030, P. R. China.

Abstract

The excessive index of a bridgeless cubic graph $G$ is the least integer $k$, such that $G$ can be covered by $k$ perfect matchings. An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless cubic graph has excessive index at most five. Clearly, Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5, so Fouquet and Vanherpe asked whether Petersen graph is the only one with that property. H\"{a}gglund gave a negative answer to their question by constructing two graphs Blowup$(K_4, C)$ and Blowup$(Prism, C_4)$. Based on the first graph, Esperet et al. constructed infinite families of cyclically 4-edge-connected snarks with excessive index at least five. Based on these two graphs, we construct infinite families of cyclically 4-edge-connected snarks $E_{0,1,2,\ldots, (k-1)}$ in which $E_{0,1,2}$ is Esperet et al.'s construction. In this note, we prove that $E_{0,1,2,3}$ has excessive index at least five, which gives a strongly negative answer to Fouquet and Vanherpe's question. As a subcase of Fulkerson conjecture, H\"{a}ggkvist conjectured that every cubic hypohamiltonian graph has a Fulkerson-cover. Motivated by a related result due to Hou et al.'s, in this note we prove that Fulkerson conjecture holds on some families of bridgeless cubic graphs.

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