An extremal problem on potentially $K_{m}-C_{4}$-graphic sequences

TL;DR

This paper investigates the minimum degree sum needed in degree sequences to guarantee the existence of a specific subgraph, the $K_m - C_4$, and proves a lower bound along with a conjecture for exact values.

Contribution

It establishes a lower bound for the degree sum ensuring potential $K_m - C_4$-graphical sequences and confirms the conjecture for the case when m=5.

Findings

Proved a lower bound for $\sigma(K_m - C_4, n)$.

Conjectured the bound is tight for all m ≥ 4.

Confirmed the conjecture for m=5.

Abstract

A sequence $S$ is potentially $K_{m}-C_{4}$-graphical if it has a realization containing a $K_{m}-C_{4}$ as a subgraph. Let $\sigma(K_{m}-C_{4}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-C_{4}, n)$ is potentially $K_{m}-C_{4}$-graphical. In this paper, we prove that $\sigma (K_{m}-C_{4}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq 4.$ We conjecture that equality holds for $n \geq m \geq 4.$ We prove that this conjecture is true for $m=5$.