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Heavy cycles passing through some specified vertices in weighted graphs

✍ Scribed by Jun Fujisawa; Kiyoshi Yoshimoto; Shenggui Zhang

Publisher: John Wiley and Sons
Year: 2005
Tongue: English
Weight: 111 KB
Volume: 49
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20066

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✦ Synopsis

Abstract

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2__d__ passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory

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