On the Toughness of the Middle Graph of a Graph

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

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

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

The middle graph \(M(G)\) of a graph \(G\) is the graph obtained from \(G\) by inserting a new vertex into every edge of \(G\) and by joining by edges those pairs of these new vertices which lie on adjacent edges of \(G\).

In this article, we give the toughness of the middle graph of a graph, and using this result we also give a sufficient condition for the middle graph to have a \(k\)-factor.