An Implicit \(\sigma_3\) Type Condition for Heavy Cycles in Weighted Graphs

Abstract

In 2003, Li introduced the concept of implicit weighted degree, denoted by \(id^w(v)\) for a vertex \(v\) in a weighted graph. In this paper, we prove that: Let \(G\) be a 2-connected weighted graph satisfying: (a) the implicit weighted degree sum of any three independent vertices is at least \(m\); (b) for each induced claw, modified claw, and FP, all edges have the same weight. Then \(G\) contains either a hamiltonian cycle or a cycle of weight at least \(\frac{2}{3}m\).