科研之友
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摘要
A graph G is 1-Hamilton-connected if G - x is Hamilton-connected for every vertex x is an element of V (G). In the article, we introduce a closure concept for 1-Hamilton-connectedness in claw-free graphs. If (G) over bar is a (new) closure of a claw-free graph G, then (G) over bar is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, (G) over bar is the line graph of a multigraph, and for some x is an element of V (G), (G) over bar - x 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.
- 出版日期2014-4
