Pairs of forbidden subgraphs and 2-connected supereulerian graphs

• Published in: Discrete mathematics Vol. 341; no. 6; pp. 1696 - 1707
• Main Authors: Čada, Roman, Ozeki, Kenta, Xiong, Liming, Yoshimoto, Kiyoshi
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.06.2018
• Subjects: Claw-free; Forbidden subgraph; Hamiltonian cycle; Supereulerian; Forbidden subgraph; Supereulerian; Hamiltonian cycle; Claw-free
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2018.03.009

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.