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Authors: R. J. Faudree and R. H. Schelp
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Authors: R. Faudree, R. Gould, M. Jacobson, L. Lesniak and T. Lindquester


It is known that if a 2-connected graphG of sufficiently large ordern satisfies the property that the union of the neighborhoods of each pair of vertices has cardinality at leastn/2, thenG is hamiltonian. In this paper, we obtain a similar generalization of Dirac’s Theorem forK(1,3)-free graphs. In particular, we show that ifG is a 2-connectedK(1,3)-free graph of ordern with the cardinality of the union of the neighborhoods of each pair of vertices at least (n+1)/3, thenG is hamiltonian. We also investigate several other related properties inK(1,3)-free graphs such as traceability, hamiltonian-connectedness, and pancyclicity.

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Authors: P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp

Let denote the class of all graphsG which satisfyG→(G 1,G 2). As a way of measuring minimality for members of, we define thesize Ramsey number ř(G 1,G 2) by.

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