Graphs with degree sequence {(m−1)m,(n−1)n} and {mn,nm}

Abstract

In this paper we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class ${\overline{G}}_{m,n}$ of the complements of such graphs. We establish various properties of $G_{m,n}$ and ${\overline{G}}_{m,n}$ related to recognition, connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We also show that several classical optimization problems on these graphs are NP-hard, while others can be solved efficiently.