Spanning eulerian subgraphs in \(N^2\)-locally connected claw-free graphs. (English) Zbl 1240.05171

Lai, Hong-Jian; Li, Mingchu; Shao, Yehong; Xiong, Liming

Ars Comb. 94, 191-199 (2010).

Summary: A graph \(G\) is \(N^{m}\)-locally connected if for every vertex \(v\) in \(G\), the vertices not equal to \(v\) and with distance at most \(m\) to \(v\) induce a connected subgraph in \(G\). In this note, we first present a counterexample to the conjecture that every 3-connected, \(N^2\)-locally connected claw-free graph is hamiltonian and then show that both connected \(N^2\)-locally connected claw-free graph and connected \(N^2\)-locally connected claw-free graph with minimum degree at least three have connected even \([2,4]\)-factors.

MSC:

05C40	Connectivity
05C45	Eulerian and Hamiltonian graphs

Keywords:

locally connected graph; claw-free graph; Hamiltonian graph; supereulerian graph