Hamiltonicity of complements of middle graphs (Q870983)

Let \(G(V,E)\) be an undirected finite simple graph. The middle graph \(M(G)\) of \(G\) has the vertex set \(V(G)\cup E(G)\) and two vertices \(x,y\) of \(M(G)\) are adjacent in \(M(G)\) if at least one of them corresponds to an edge \(e\) of \(G\) and the other one is either one of the endvertices of \(e\) in \(G\), or corresponds to an edge \(f\) of \(G\) adjacent to \(e\) in \(G\). It is shown that the complement of the middle graph \(M(G)\) is Hamiltonian if and only if \(G\) is not a star and is not isomorphic to one of the graphs \(K_1, 2K_1, K_2, K_1\cup K_2, K_3, K_1\cup K_3\).