A note on a degree sum condition for long cycles in graphs

TL;DR

This paper proposes a conjecture extending Ore's degree sum condition for Hamiltonicity to long cycles in 2-connected graphs, and verifies it for the case when k=1, suggesting a broader understanding of cycle lengths based on degree sums.

Contribution

It introduces a new conjecture generalizing Ore's condition for long cycles and proves it for the specific case when k=1.

Findings

Conjecture holds for k=1.

Generalizes Ore's degree sum condition.

Provides insight into cycle lengths in 2-connected graphs.

Abstract

We conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\geq n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n/2$), unless $G$ is bipartite and $n-k$ is odd. This generalizes to long cycles a well-known degree sum condition for hamiltonicity of Ore. The conjecture is shown to hold for $k=1$.