Tight Toughness and Isolated Toughness for $\{K_2,C_n\}$-factor critical avoidable graph (2406.17631v1)

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.