[Paper Review] An extremal problem on potentially $K_{m}-C_{4}$-graphic sequences

Chunhui Lai|ArXiv.org|Sep 3, 2004

Digital Image Processing Techniques8 references3 citations

TL;DR

This paper investigates the extremal problem of determining the minimum degree sum ensuring that a graphical sequence contains a $K_m - C_4$ subgraph. It proves a lower bound for $\sigma(K_m - C_4, n)$ and confirms equality for $m=5$, establishing $\sigma(K_5 - C_4, n) = 4n - 4$ for $n \geq 5$, supporting a broader conjecture on the exact threshold for all $n \geq m \geq 4$. The result extends extremal graph theory to potentially $H$-graphic sequences with forbidden substructures.

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 $σ(K_{m}-C_{4}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-C_{4}, n)$ is potentially $K_{m}-C_{4}$-graphical. In this paper, we prove that $σ(K_{m}-C_{4}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ 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 $m=5$.

Motivation & Objective

• To determine the smallest degree sum $\sigma(K_m - C_4, n)$ such that every $n$-term graphical sequence with that sum is potentially $K_m - C_4$-graphic.
• To establish a lower bound for $\sigma(K_m - C_4, n)$ for $n \geq m \geq 4$.
• To prove that the conjectured exact formula for $\sigma(K_m - C_4, n)$ holds when $m = 5$.
• To extend extremal graph theory results from forcibly $H$-graphic sequences to potentially $H$-graphic sequences involving $K_m - C_4$ subgraphs.

Proposed method

• Constructs a graph $H = K_{m-3} + \overline{K_{n-m+3}}$ as a non-$K_m - C_4$-containing realization to derive a lower bound on $\sigma(K_m - C_4, n)$.
• Applies induction on $n$ to prove that all $n$-term graphical sequences with $\sigma(S) \geq 4n - 4$ are potentially $K_5 - C_4$-graphic for $n \geq 5$.
• Uses degree sequence analysis and vertex removal to reduce the problem to smaller cases, leveraging known results on $K_4$-containing realizations.
• Employs edge-switching techniques to modify graph realizations and construct a $K_5 - C_4$ subgraph when necessary.
• Applies results from prior works (e.g., [4], [6]) on $K_4$-containing realizations and degree sequence properties to support the inductive argument.
• Uses the fact that $K_5 - C_4$ is formed by removing four edges of a 4-cycle from $K_5$, and constructs such a subgraph through strategic vertex and edge analysis.

Experimental results

Research questions

• RQ1 What is the minimum degree sum $\sigma(K_m - C_4, n)$ that guarantees a graphical sequence is potentially $K_m - C_4$-graphic for $n \geq m \geq 4$?
• RQ2 Does the lower bound $\sigma(K_m - C_4, n) \geq (2m - 6)n - (m - 3)(m - 2) + 2$ hold with equality for all $n \geq m \geq 4$?
• RQ3 Is the conjectured formula $\sigma(K_m - C_4, n) = (2m - 6)n - (m - 3)(m - 2) + 2$ valid for $m = 5$?
• RQ4 Can the extremal threshold for potentially $K_m - C_4$-graphic sequences be characterized via degree sum and inductive construction?

Key findings

• The paper establishes a lower bound: $\sigma(K_m - C_4, n) \geq (2m - 6)n - (m - 3)(m - 2) + 2$ for all $n \geq m \geq 4$.
• For $m = 5$, the exact value is proven: $\sigma(K_5 - C_4, n) = 4n - 4$ for all $n \geq 5$.
• The construction of a graph $H = K_{m-3} + \overline{K_{n-m+3}}$ with no $K_m - C_4$ subgraph confirms the lower bound via degree sum calculation.
• The inductive proof for $m = 5$ relies on analyzing vertex degrees and using edge-switching to construct the required $K_5 - C_4$ subgraph.
• The result confirms the conjecture for $m = 5$, supporting the broader conjecture that $\sigma(K_m - C_4, n) = (2m - 6)n - (m - 3)(m - 2) + 2$ holds for all $n \geq m \geq 4$.
• The proof shows that any $n$-term graphical sequence with $\sigma(S) \geq 4n - 4$ admits a realization containing $K_5 - C_4$, even under degree constraints like $d_n \geq 3$.