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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