Non-Hamilton cycle sets of having solutions and their properties

TL;DR

This paper explores the structure of specific cycle sets in reduced graphs, focusing on their properties and conditions under which they exhibit Hamiltonicity, thereby advancing understanding of cycle configurations in graph theory.

Contribution

It characterizes the cycle structure of 2-common (v, 0) combinations in reduced graphs and provides conditions for their Hamiltonicity, a novel analysis in graph cycle theory.

Findings

Characterization of 2-common (v, 0) cycle sets in reduced graphs

Conditions determining Hamiltonicity of these cycle sets

Insights into cycle structure and graph reduction techniques

Abstract

A graph \textit{G} is a tuple (\textit{V}, \textit{E}), where \textit{V} is the vertex set, \textit{E} is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an edge except for leaving only one vertex of degree 2. A 2-common (\textit{v}, \textit{0}) combination is a cycle set in which every pair of joint cycles \textit{A} and \textit{B} satisfies $|V(A)\cap V(B)|=2$ and $|E(A)\cap E(B)|=0$. In this paper, we investigate the cycle structure of 2-common (\textit{v}, \textit{0}) combination in reduced graphs, and give the characterizations of their Hamiltoncity.