Snarks, Hypohamiltonian Graphs and Non-Supereulerian Graphs

A graph G is hypohamiltonian if it is not Hamiltonian but for each v. V(G), the graph G - v is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset R subset of V(G), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76: 173-181, 1990] and in [J. Combin. Theory, Ser B 66: 123-139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2): 291-304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Universite Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.