On degree sets and the minimum orders in bipartite graphs
Yannis Manoussakis, H.P. Patil · Discussiones Mathematicae Graph Theory · 2014
For any simple graph G, let D(G) denote the degree set {deg G (v) : v ∈ V (G)}.Let S be a finite, nonempty set of positive integers.In this paper, we first determine the families of graphs G which are unicyclic, bipartite satisfying D(G) = S, and further obtain the graphs of minimum orders in such families.More general, for a given pair (S, T ) of finite, nonempty sets of positive integers of the same cardinality, it is shown that there exists a bipartite graph B(X, Y ) such that D(X) = S, D(Y ) = T and the minimum orders of different types are obtained for such graphs.