A Characterization for a Graphic Sequence to be Potentially $K_{1^3},4$-Graphic

Abstract

A graphic sequence $\pi =(d_1,d_2,\dots ,d_n)$ is said to be potentially $K_{1^3,4}$-graphic if there is a realization of $\pi$ containing $K_{1^3,4}$ as a subgraph, where $K_{1^3,4}$ is the $1\times 1\times 1\times 4$ complete 4-partite graph. In this paper, we characterize the graphic sequences potentially $K_{1^3,4}$-graphic and the result is simple. In addition, we apply this characterization to compute the values of $\sigma (K_{1^3,4},n)$.