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(K4‎,‎C) and Blowup(Prism‎,‎C4)‎. ‎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 E0,1,2,…‎,‎(k−1) in which E0,1,2 is Esperet et al. s construction‎. ‎In this note‎, ‎we prove that E0,1,2,3 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.