A Note for Integer Sequences to be Potentially $K_{r+I}$-Graphic

Abstract

Let $K_{r+1}$ be the complete graph on $r+1$ vertices and let $\pi =(d_1,d_2,\dots ,d_n)$ be a non-increasing sequence of nonnegative integers. If $\pi$ has a realization containing $K_{r+1}$ as a subgraph, then $\pi$ is said to be potentially $K_{r+1}$-graphic. A.R. Rao obtained an Erdős-Gallai type criterion for $\pi$ to be potentially $K_{r+1}$-graphic. In this paper, we provide a simplification of this Erdős-Gallai type criterion. Additionally, we present the Fulkerson-Hoffman-McAndrew type criterion and the Hasselbarth type criterion for $\pi$ to be potentially $K_{r+1}$-graphic.