Tight Toughness and Isolated Toughness for $\{K_2,C_n\}$-factor critical avoidable graph

TL;DR

This paper establishes new sufficient conditions based on toughness measures for graphs to be $ ext{K}_2$- and $C_n$-factor critical avoidable, advancing understanding of graph resilience and factorization properties.

Contribution

It introduces novel toughness-based criteria for identifying $ ext{K}_2$- and $C_n$-factor critical avoidable graphs, extending previous graph factorization theory.

Findings

Derived sufficient conditions involving isolated toughness for $ ext{K}_2,C_n$-factor critical avoidable graphs.

Established criteria combining tight toughness and isolated toughness for specific cycle lengths.

Enhanced theoretical framework for graph resilience and factorization properties.

Abstract

A spannning subgraph $F$ of $G$ is a $\{K_2,C_n\}$-factor if each component of $F$ is either $K_{2}$ or $C_{n}$. A graph $G$ is called a $(\{K_2,C_n\},n)$-factor critical avoidable graph if $G-X-e$ has a $\{K_2,C_n\}$-factor for any $S\subseteq V(G)$ with $|X|=n$ and $e\in E(G-X)$. In this paper, we first obtain a sufficient condition with regard to isolated toughness of a graph $G$ such that $G$ is $\{K_2,C_{n}\}$-factor critical avoidable. In addition, we give a sufficient condition with regard to tight toughness and isolated toughness of a graph $G$ such that $G$ is $\{K_2,C_{2i+1}|i \geqslant 2\}$-factor critical avoidable respectively.