Abstract
As an extension of quasi claw-free graphs, the class of P 3-dominated graphs has been introduced by Broersma and Vumar (Math Methods Oper Res 69:297–306, 2009). For a noncomplete graph G, the number NC and NC 2 are defined as \({NC=\min\{|N(x)\cup N(y)|: x,y\in V(G) {\rm and} xy\notin E(G)\}\, {\rm and} NC_2=\min\{|N(x)\cup N(y)|: x,y\in V(G)\, {\rm and}\, d(x,y)=2 \}}\), respectively. For a complete graph G, set \({NC=NC_{2}=|V(G)|-1}\). In this paper, we prove that a 2-connected P 3-dominated graph of order n is traceable if \({NC\geq (n-2)/2}\). Moreover, we prove that a 3-connected P 3-dominated graph of order n is hamiltonian if \({NC_2\geq (2n-6)/3}\). Our results extend some previous results on claw-free graphs.
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This work is supported by NSFC (No.11061034), XJEDU2010I01 and the Natural Science Foundation of Xinjiang University (XY110102).
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Ma, X., Vumar, E. Neighborhood Union Conditions for Hamiltonicity of P 3-Dominated Graphs. Graphs and Combinatorics 30, 1499–1511 (2014). https://doi.org/10.1007/s00373-013-1354-4
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DOI: https://doi.org/10.1007/s00373-013-1354-4