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A generalization of Dirac’s theorem for \(K(1,3)\)-free graphs. (English) Zbl 0738.05054
Period. Math. Hung. (to appear).
Summary: It is known that if a \(2\)-connected graph \(G\) of sufficiently large order \(n\) satisfies the property that the union of the neighborhoods of each pair of vertices has cardinality at least \({n/2}\), then \(G\) is hamiltonian. In this paper, we obtain a similar generalization of Dirac’s theorem for \(K(1,3)\)-free graphs. In particular, we show that if \(G\) is a 2-connected \(K(1,3)\)-free graph of order \(n\) with the cardinality of the union of the neighborhoods of each pair of vertices at least \({(n+1)\over 3}\), then \(G\) is hamiltonian. We also investigate several other related properties in \(K(1,3)\)-free graphs such as traceability, hamiltonian- connectedness, and pancyclicity.

MSC:
05C45 Eulerian and Hamiltonian graphs
05C40 Connectivity