Pairs of forbidden subgraphs and 2-connected supereulerian graphs

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, Discrete Math. 332 (2014) 15-22], 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 ∗Research partially supported by project P202/12/G061 of the Czech Science Foundation and by the European Regional Development Fund (ERDF), project NTIS New Technologies for Information Society, European Centre of Excellence, CZ.1.05/1.1.00/02.0090 †This work was supported by JST ERATO Grant Number JPMJER1201, Japan. ‡Supported by Nature Science Funds of China (No. 11471037 and No. 11671037) §Research supported by JSPS KAKENHI Grant Number 26400190