A note on Fouquet-Vanherpe’s question and Fulkerson conjecture

‎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.