Pairs of forbidden subgraphs and 2-connected supereulerian graphs

Abstract

Let G be a 2-connected claw-free graph. We show that •if G is N1,1,4-free or N1,2,2-free or Z5-free or P8-free, respectively, then G has a spanning Eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs,•if the minimum degree δ(G)≥3 and G is N2,2,5-free or Z9-free, respectively, then G has a spanning Eulerian subgraph or its closure is the line graph of a graph in a family of well-defined graphs.Here Zi (Ni,j,k) denotes the graph obtained by attaching a path of length i≥1 (three vertex-disjoint paths of lengths i,j,k≥1, respectively) to a triangle.Combining our results with a result in [Xiong (2014)], we prove that all 2-connected hourglass-free claw-free graphs G with one of the same forbidden subgraphs above (or additionally δ(G)≥3) are hamiltonian with the same excluded families of graphs. In particular, we prove that every 3-edge-connected claw-free hourglass-free graph that is N2,2,5-free or Z9-free is hamiltonian.