Hypohamiltonian Snarks Have a 5-Flow

Breno Freitas, Cândida Nunes da Silva, Cláudio L. Lucchesi

Abstract: It is well known that a snark does not admit a 3-edge colouring, neither a 4-flow, nor a Hamiltonian cycle. A snark is 4-edge-(flow)-critical if the contraction of any of its edges yields a graph that has a 4-flow; it is 2-vertex critical if the removal of any two adjacent vertices yields a graph that has a 3-edge-colouring; and hypohamiltonian if the removal of any of its vertices yields a Hamiltonian graph. In this paper we show that a snark is 4-edge-critical if and only if it is 2-vertex-critical and also that every hypohamiltonian snark admits a 5-flow, thus providing an answer to a question proposed by Cavicchioli et al. in 2003.