On Pure Degree Sequence Manipulations Forcing Long Cycles in Graphs

TL;DR

This paper introduces new pure degree sequence manipulation techniques that simplify the derivation of conditions for long cycles in graphs, expanding the theoretical framework for Hamiltonian cycle criteria.

Contribution

It presents two novel types of pure degree manipulations and their long-cycle counterparts, offering simpler and more verifiable results in graph theory.

Findings

New degree manipulation methods for long cycles

Simpler verification of Hamiltonian conditions

Extensions to existing degree sequence criteria

Abstract

The well-known Hamiltonian sufficient conditions, proposed by Dirac, Faudree et al., P\'osa, Bondy, Chv\'atal are based on pure degree manipulations without any additional conditions. In this paper, we present two new types of pure degree manipulations that produce relatively simpler (in view of verification) results. The reverse versions (long-cycle versions) of the obtained results are presented as well. The importance of pure degree manipulations is motivated by their amenability (as starting points) to generalizations and new ideas in a great variety of ways.