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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
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