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

The excessive index of a bridgeless cubic graph G is the least integer ký, ýsuch that G can be covered by k perfect matchingsý. ýAn equivalent form of Fulkerson conjecture (due to Berge) is that every bridgelessý ýcubic graph has excessive index at most fiveý. ýClearlyý, ýPetersen graph is a cyclically 4-edge-connected snark with excessive index at least 5ý, ýso Fouquet and Vanherpe asked whether Petersen graph is the only one with that propertyý. ýH\"{a}gglund gave a negative answer to their question by constructing two graphs Blowup(K4ý,ýC) and Blowup(Prismý,ýC4)ý. ýBased on the first graphý, ýEsperet et alý. ýconstructed infinite families of cyclically 4-edge-connected snarks with excessive index at least fiveý. ýBased on these two graphsý, ýwe construct infinite families of cyclically 4-edge-connected snarks E0,1,2,…ý,ý(k−1) in which E0,1,2 is Esperet et al.'s constructioný. ýIn this noteý, ýwe prove that E0,1,2,3 has excessive index at least fiveý, ýwhich gives a strongly negative answer to Fouquet and Vanherpe's questioný. ýAs a subcase of Fulkerson conjectureý, ýH\"{a}ggkvist conjectured that every cubic hypohamiltonian graph has a Fulkerson-coverý. ýMotivated by a related result due to Hou et al.'sý, ýin this note we prove that Fulkerson conjecture holds on some families of bridgeless cubic graphs.