An Weighted Implicit Degree Condition
for Heavy
Cycles in Weighted Graphs
Bing Chen and Shenggui
Zhang
small Department of Applied Mathematics,
Northwestern Polytechnical University, Xi'an 710072,
P.R. China
Abstract
A weighted graph is one in which
every edge e is assigned a nonnegative number w(e), called the
weight of e. The weight of a cycle is defined as the sum of the
weights of its edges. The weighted degree of a vertex is the sum
of the weights of the edges incident with it. In a recent paper of
Li, the author introdcued the concept of weighted implicit degree
of vertices in weighted graphs. This notion generalizes the
concept of implicit degree of vertices in unweighted graphs which
was first proposed by Zhu et al. It was shown that the weighted
implicit degree of a vertex in a weighted graph is no less than
that of the weighted degree of the vertex. We use idw(v) for
the weighted implicit degree of a vertex v. In this paper, we
prove that a 2-connected weighted graph G contains either a
Hamilton cycle or a cycle of weight at least m if it satisfies
the following conditions: (1) max 
for each pair of nonadjacent vertices u and v that are
vertices of an induced claw or an induced modified claw of G;
(2) In each induced claw, each induced modified claw and each
induced P4 of G, all edges have the same weight. This is a
common generalization for several results on the existence of long
cycles in unweighted graphs and that of heavy cycles in weighted
graphs.
