Potentially $K_{m}-G$-graphical Sequences: A Survey

TL;DR

This survey reviews recent research on potentially $K_{m}-G$-graphic sequences, focusing on conditions for sequences to contain specific subgraphs and classifying degree sum thresholds for such properties.

Contribution

It provides a comprehensive summary of recent results and introduces a classification method for determining degree sum thresholds for potentially $K_{m}-G$-graphic sequences.

Findings

Summarizes recent advances in potentially $K_{m}-G$-graphic sequences

Provides a classification for $\sigma(H,n)$ related to degree sums

Highlights key conditions for subgraph containment in degree sequences

Abstract

The set of all non-increasing nonnegative integers sequence $\pi=$ ($d(v_1),$ $d(v_2),$ $...,$ $d(v_n)$) is denoted by $NS_n$. A sequence $\pi\in NS_n$ 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 $NS_n$ is denoted by $GS_n$. 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 $\sigma(H,n)$.