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.
A Closure for 1-Hamilton-Connectedness in Claw-Free Graphs
Article first published online: 8 MAY 2013
© 2013 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 75, Issue 4, pages 358–376, April 2014
How to Cite
Ryjáček, Z. and Vrána, P. (2014), A Closure for 1-Hamilton-Connectedness in Claw-Free Graphs. J. Graph Theory, 75: 358–376. doi: 10.1002/jgt.21743
- Issue published online: 15 JAN 2014
- Article first published online: 8 MAY 2013
- Manuscript Revised: 4 APR 2013
- Manuscript Received: 22 JUN 2012
- Czech Science Foundation and by the European Regional Development Fund (ERDF). Grant Number: P202/12/G061
- NTIS - New Technologies for Information Society, European Centre of Excellence. Grant Number: CZ.1.05/1.1.00/02.0090
- claw-free graph;
- line graph;
- Thomassen Conjecture
A graph G is 1-Hamilton-connected if is Hamilton-connected for every vertex . In the article, we introduce a closure concept for 1-Hamilton-connectedness in claw-free graphs. If is a (new) closure of a claw-free graph G, then is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, is the line graph of a multigraph, and for some , 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.