On the Toughness of the Total Graph of a Graph

Abstract

The toughness $t(G)$ of a noncomplete graph $G$ is defined as

$$t(G)=min\{\frac{|S|}{\omega (G–S)}\mid S\subset V(G),\omega (G–S)\ge 2\},$$

where $\omega (G–S)$ is the number of components of $G–S$. We also define $t(K_n)=+\mathrm{\infty }$ for every $n$.

The total graph $T(G)$ of a graph $G$ is the graph whose vertex set can be put in one-to-one correspondence with the set $V(G)\cup E(G)$ such that two vertices of $T(G)$ are adjacent if and only if the corresponding elements of $G$ are adjacent or incident.

In this article, we study the toughness of the total graph $T(G)$ of a graph $G$ on at least $3$ vertices and give especially that $t(T(G))=t(G)$ if $\kappa (G)=\lambda (G)$ and $\kappa (G)\le 2$, where $\kappa (G)$ and $\lambda (G)$ are the vertex and the edge-connectivity of $G$, respectively.