On the Toughness of the Total Graph of a Graph

Abstract

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

\[t(G) = \min \left\{ \frac{|S|}{\omega(G – S)} \mid S \subset V(G), \omega(G – S) \geq 2 \right\},\]

where \(\omega(G – S)\) is the number of components of \(G – S\). We also define \(t(K_n) = +\infty\) 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 \(\kappa(G) \leq 2\), where \(\kappa(G)\) and \(\lambda(G)\) are the vertex and the edge-connectivity of \(G\), respectively.