The Smallest Degree Sum that Yields Potentially \(K_{2,s}\)-graphic Sequences

Abstract

Let \(\sigma(K_{r,s}, n)\) denote the smallest even integer such that every \(n\)-term positive graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with term sum \(\sigma(\pi) = d_1 + d_2 + \cdots + d_n \geq \sigma(K_{r,s}, n)\) has a realization \(G\) containing \(K_{r,s}\) as a subgraph, where \(K_{r,s}\) is the \(r \times s\) complete bipartite graph. In this paper, we determine \(\sigma(K_{2,3}, n)\) for \(m \geq 5\). In addition, we also determine the values \(\sigma(K_{2,s}, n)\) for \(s \geq 4\) and \(n \geq 2[\frac{(s+3)^2}{4}]+5\).