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 $\pi =(d_1,d_2,\dots ,d_n)$ with term sum $\sigma (\pi )=d_1+d_2+\cdots +d_n\ge \sigma (K_{r,s},n)$ 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 $m\ge 5$. In addition, we also determine the values $\sigma (K_{2,s},n)$ for $s\ge 4$ and $n\ge 2[\frac{(s+3{)}^2}{4}]+5$.