Pairs of forbidden subgraphs and 2-connected supereulerian graphs

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 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 N_2,2,5-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 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 Z₉-free is hamiltonian.