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Induced hourglass and the equivalence between Hamiltonicity and supereulerianity in claw-free graphs. (English) Zbl 1298.05193
Summary: A graph $$H$$ has the hourglass property if in every induced hourglass $$S$$ (the unique simple graph with the degree sequence (4, 2, 2, 2, 2)), there are two non-adjacent vertices which have a common neighbor in $$H - V(S)$$. Let $$G$$ be a claw-free simple graph and $$k$$ a positive integer. In this paper, we prove that if either $$G$$ is hourglass-free or $$G$$ has the hourglass property and $$\delta(G) \geq 4$$, then $$G$$ has a 2-factor with at most $$k$$ components if and only if it has an even factor with at most $$k$$ components. We provide some of its applications: combining the result (the case when $$k = 1$$) with [F. Jaeger, J. Graph Theory 3, 91–93 (1979; Zbl 0396.05034); Z.-H. Chen et al., J. Comb. Math. Comb. Comput. 59, 165–171 (2006; Zbl 1124.05054)], we obtain that every 4-edge-connected claw-free graph with the hourglass property is Hamiltonian and that every essentially 4-edge-connected claw-free hourglass-free graph of minimum degree at least three is Hamiltonian, thereby generalizing the main result in [T. Kaiser et al., J. Graph Theory 48, No. 4, 267–276 (2005; Zbl 1060.05064)] and the result in [H. J. Broersma et al., J. Graph Theory 37, No. 2, 125–136 (2001; Zbl 0984.05067)] respectively in which the conditions on the vertex-connectivity are replaced by the condition of (essential) 4-edge-connectivity. Combining our result with [P. A. Catlin and H.-J. Lai, Ars Comb. 30, 177–191 (1990; Zbl 0751.05064); H.-J. Lai et al., Ars Comb. 94, 191–199 (2010; Zbl 1240.05171); P. Paulraja, Ars Comb. 24, 57–65 (1987; Zbl 0662.05044)], we also obtain several other results on the existence of a Hamiltonian cycle in claw-free graphs in this paper.

##### MSC:
 05C45 Eulerian and Hamiltonian graphs 05C38 Paths and cycles
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##### References:
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