Hamiltonicity of complements of middle graphs

Abstract

For a graph G, the middle graph M ( G ) of G is the graph with vertex set V ( G ) ∪ E ( G ) in which the vertices x and y are joined by an edge if { x , y } ∩ E ( G ) ≠ ∅ , and x and y are adjacent or incident in G. In this note, we show that the complement of middle graph M ( G ) of a graph G is hamiltonian if and only if G is not a star and is not isomorphic to any graph in { K 1 , 2 K 1 , K 2 , K 2 ∪ K 1 , K 3 , K 3 ∪ K 1 } .