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On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance
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On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance
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On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance
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Journal Article
On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance
2016
Overview
Assume that$n, \\delta ,k$are integers with$0 \\leq k < \\delta < n$ . Given a graph$G=(V,E)$with$|V|=n$ . The symbol$G-F, F \\subseteq V$ , denotes the graph with$V(G-F)=V-F$ , and$E(G-F)$obtained by$E$after deleting the edges with at least one endvertex in$F$ .$G$is called$k$ -vertex fault traceable,$k$ -vertex fault Hamiltonian, or$k$ -vertex fault Hamiltonian-connected if$G-F$remains traceable, Hamiltonian, and Hamiltonian-connected for all$F$with$0 \\leq |F| \\leq k$ , respectively. The notations$h_1(n, \\delta ,k)$ ,$h_2(n, \\delta ,k)$ , and$h_3(n, \\delta ,k)$denote the minimum number of edges required to guarantee an$n$ -vertex graph with minimum degree$\\delta (G) \\geq \\delta$to be$k$ -vertex fault traceable,$k$ -vertex fault Hamiltonian, and$k$ -vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the$k$ -vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for$h_i(n, \\delta ,k)$for$1 \\leq i \\leq 3$ , which improves and extends the known results for$k=0$ .
Publisher
DMTCS,Discrete Mathematics & Theoretical Computer Science
