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A Closure for 1-Hamilton-Connectedness in Claw-Free Graphs

Authors

  • Zdeněk Ryjáček,

    1. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WEST BOHEMIA, CZECH REPUBLIC
    2. INSTITUTE FOR THEORETICAL COMPUTER SCIENCE (ITI), CHARLES UNIVERSITY, CZECH REPUBLIC
    3. SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE, THE UNIVERSITY OF NEWCASTLE, AUSTRALIA
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  • Petr Vrána

    1. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WEST BOHEMIA, CZECH REPUBLIC
    2. INSTITUTE FOR THEORETICAL COMPUTER SCIENCE (ITI), CHARLES UNIVERSITY, CZECH REPUBLIC
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  • Research supported by project P202/12/G061 of the Czech Science Foundation and by the European Regional Development Fund (ERDF), project NTIS - New Technologies for Information Society, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.

Abstract

A graph G is 1-Hamilton-connected if math formula is Hamilton-connected for every vertex math formula. In the article, we introduce a closure concept for 1-Hamilton-connectedness in claw-free graphs. If math formula is a (new) closure of a claw-free graph G, then math formula is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, math formula is the line graph of a multigraph, and for some math formula, math formula is the line graph of a multigraph with at most two triangles or at most one double edge. As applications, we prove that Thomassen's Conjecture (every 4-connected line graph is hamiltonian) is equivalent to the statement that every 4-connected claw-free graph is 1-Hamilton-connected, and we present results showing that every 5-connected claw-free graph with minimum degree at least 6 is 1-Hamilton-connected and that every 4-connected claw-free and hourglass-free graph is 1-Hamilton-connected.

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