Elsevier

Discrete Mathematics

Volume 307, Issue 1, 6 January 2007, Pages 38-53
Discrete Mathematics

Heavy fans, cycles and paths in weighted graphs of large connectivity

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Abstract

A set of paths joining a vertex y and a vertex set L is called (y,L)-fan if any two of the paths have only y in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices.

In this paper, we show the existence of heavy fans with large width containing some specified vertices in weighted graphs of large connectivity, which is a weighted analogue of Perfect's theorem. Using this, in 3-connected weighted graphs, we can find heavy cycles containing three specified vertices, and also heavy paths joining two specified vertices containing two more specified vertices. These results extend the previous results in 2-connected weighted graphs to 3-connected weighted graphs.

Keywords

Weighted graph
Perfect's theorem
Heavy cycle
Heavy path
Specified vertex

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1

This work is partially done when the author was visiting University of Twente.