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 N 1 , 1 , 4 -free or N 1 , 2 , 2 -free or Z 5 -free or P 8 -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 N 2 , 2 , 5 -free or Z 9 -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 Z i ( N i , 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 N 2 , 2 , 5 -free or Z 9 -free is hamiltonian.