Potentially $K_m-G$-graphical Sequences: A Survey

Abstract

The set of all non-increasing nonnegative integers sequence $\pi =$ ($d(v_1),$ $d(v_2),$ $...,$ $d(v_n)$) is denoted by $N{S}_n$. A sequence $\pi \in N{S}_n$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $\pi$. The set of all graphic sequences in $N{S}_n$ is denoted by $G{S}_n$. A graphical sequence $\pi$ is potentially $H$-graphical if there is a realization of $\pi$ containing $H$ as a subgraph, while $\pi$ is forcibly $H$-graphical if every realization of $\pi$ contains $H$ as a subgraph. Let $K_k$ denote a complete graph on $k$ vertices. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_m$). This paper summarizes briefly some recent results on potentially $K_m-G$-graphic sequences and give a useful classification for determining $\sigma (H,n)$.

18 pages

References in corpus (1)

• A note on potentially $K_4-e$ graphical sequences