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  • 1 Memphis State University Dept. of Mathematical Sciences 38152 Memphis Tennessee 38152 Memphis Tennessee
  • 2 Emony University Department of Mathematics and Computer Science 30322 Atlanta Georgia 30322 Atlanta Georgia
  • 3 University of Louisville Department of Mathematics 40292 Louisville Kentucky 40292 Louisville Kentucky
  • 4 Drew University Department of Mathematics and Computer Science 02940 Madison NJ 02940 Madison NJ
  • 5 Rhodes College Department of Mathematics 38112 Memphis Tennessee 38112 Memphis Tennessee
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Abstract  

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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