Heavy subgraph conditions for longest cycles to be heavy in graphs

TL;DR

This paper characterizes graphs that lack heavy cycles and identifies conditions under which longest cycles in 2-connected graphs are heavy, based on heavy subgraph conditions involving degree sums.

Contribution

It introduces a characterization of graphs with no heavy cycles and determines specific heavy subgraph conditions ensuring longest cycles are heavy.

Findings

Characterized graphs with no heavy cycles.

Identified all connected graphs S where S-heavy 2-connected graphs have heavy longest cycles.

Abstract

Let $G$ be a graph on $n$ vertices. A vertex of $G$ with degree at least $n/2$ is called a heavy vertex, and a cycle of $G$ which contains all the heavy vertices of $G$ is called a heavy cycle. In this paper, we characterize the graphs which contain no heavy cycles. For a given graph $H$, we say that $G$ is $H$-\emph{heavy} if every induced subgraph of $G$ isomorphic to $H$ contains two nonadjacent vertices with degree sum at least $n$. We find all the connected graphs $S$ such that a 2-connected graph $G$ being $S$-heavy implies any longest cycle of $G$ is a heavy cycle.