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 $i{d}^w(v)$ 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 $\frac{2}{3}m$.