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Degree Sum Condition for k-ordered Hamiltonian Connected Graphs

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Abstract

Let G be a graph on n vertices. If for any ordered set of vertices S = {v 1, v 2, . . . , v k }, that is, the vertices in S appear in order of the sequence v 1, v 2, . . . , v k , there exists a v 1 − v k (hamiltonian) path containing S in the given order, then G is k-ordered (hamiltonian) connected. Let {u 1, u 2} and {u 3, u 4} be distinct pairs of nonadjacent vertices. When \({G\neq{K_n}}\) and \({G\neq{K_n-e}}\) , we define \({\sigma'_4(G)={\rm min}\{d_G(u_1)+d_G(u_2)+d_G(u_3)+d_G(u_4)\}}\) , otherwise set \({\sigma_4'(G)=\infty}\) . In this paper we will present some sufficient conditions for a graph to be k-ordered connected based on \({\sigma_4'(G)}\) . As a main result we will show that if \({\sigma_4'(G)\geq{2n+3k-10}}\) \({(4 \le k \le \frac{n+1}{2})}\) , then G is k-ordered hamiltonian connected. Our outcomes generalize several related results known before.

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Authors and Affiliations

  1. Department of Mathematics, Millsaps College, Jackson, MS, 39210, USA

    Emlee W. Nicholson

  2. Department of Mathematics, University of Mississippi, University, MS, 38677, USA

    Bing Wei

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  1. Emlee W. Nicholson
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  2. Bing Wei
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Correspondence to Emlee W. Nicholson.

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Nicholson, E.W., Wei, B. Degree Sum Condition for k-ordered Hamiltonian Connected Graphs. Graphs and Combinatorics 31, 743–755 (2015). https://doi.org/10.1007/s00373-013-1393-x

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  • DOI: https://doi.org/10.1007/s00373-013-1393-x

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