Hamiltonicity of Complements of Total Graphs

Abstract

For a graph G, the total graph T(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 and y are adjacent or incident in G. In this paper, we show that the complement of total graph T(G) of a simple graph G is hamiltonian if and only if G is not isomorphic to any graph in {K1, r | r ≥ 1}∪{K1, s + K 1| s ≥ 1}∪{K1, t + e| t ≥ 2}∪{K2 + 2K1, K3 + ,K1, K3 + 2K 1, K4}. © 2007 Springer-Verlag Berlin Heidelberg.