Relative Length of Long Paths and Cycles in Graphs

TL;DR

This paper establishes a new lower bound on the length of the longest cycle in 2-connected graphs based on the length of the longest path and degree conditions, improving previous bounds by Nash-Williams, Bondy, and others.

Contribution

It introduces a novel degree sum condition involving triples of independent vertices that guarantees the cycle length is at least the path length minus one, refining existing theorems.

Findings

Proves that under certain degree sum conditions, the longest cycle length is at least the longest path length minus one.

Improves bounds relating degree sums and cycle lengths in 2-connected graphs.

Provides a new criterion for cycle length based on independent triples' degree sums.

Abstract

For a graph $G$, $n$ denotes the order of $G$, $p$ the order of a longest path in $G$ and $c$ the order of a longest cycle. We show that if $G$ is a 2-connected graph such that $d(x)+d(y)+d(z)\ge p+2$ for all triples $x,y,z$ of independent vertices, then $c\ge p-1$. This improves results of Nash-Williams (in terms of minimum degree $\delta$ and order $n$), Bondy (in terms of degree sum $\sigma_{3}$ and order $n$), and Enomoto, Heuvel, Kaneko and Saito (in terms of degree sum $\sigma_3$, order $n$ and relative length $diff(G)=p-c$).