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