[Paper Review] Making a K_4-free graph bipartite

Benny Sudakov|ArXiv.org|Jun 27, 2007

Limits and Structures in Graph Theory8 references6 citations

TL;DR

This paper proves that every $K_4$-free graph on $n$ vertices can be made bipartite by deleting at most $n^2/9$ edges, with equality only in the case of a complete 3-partite graph with parts of size $n/3$. The result confirms a longstanding conjecture by Erdős and establishes a tight extremal bound using the regularity lemma and structural analysis of $K_4$-free graphs.

ABSTRACT

We show that every K_4-free graph G with n vertices can be made bipartite by deleting at most n^2/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdos.

Motivation & Objective

• To resolve a conjecture by Paul Erdős on the maximum number of edges that need to be deleted to make a $K_4$-free graph bipartite.
• To determine the extremal graph achieving the upper bound on edge deletions in $K_4$-free graphs.
• To extend the result to $H$-free graphs with chromatic number 4 using Szemerédi's Regularity Lemma.
• To provide a tight asymptotic bound for the bipartition problem in dense $K_r$-free graphs.

Proposed method

• Apply Szemerédi's Regularity Lemma to partition the vertex set of a $K_4$-free graph into $\epsilon$-regular pairs with bounded parts.
• Use the regularity lemma to show that if a graph has many dense regular pairs, it must contain a $K_4$, contradicting the $K_4$-freeness.
• Prove that any $K_4$-free graph with more than $n^2/9$ edges must have a dense 3-partite structure.
• Use extremal graph theory to show that the only graph achieving the bound $n^2/9$ is the complete 3-partite graph with equal parts.
• Apply the main result to $H$-free graphs with $\chi(H) = 4$, using the regularity lemma to embed a complete 4-partite subgraph if the density conditions are met.
• Combine edge deletion bounds from the main theorem with density and regularity arguments to derive the asymptotic bound $ (1+o(1))n^2/9 $ for $H$-free graphs.

Experimental results

Research questions

• RQ1 What is the maximum number of edges that must be deleted to make any $K_4$-free graph on $n$ vertices bipartite?
• RQ2 Is the complete 3-partite graph with parts of size $n/3$ the only extremal graph requiring $n^2/9$ edge deletions?
• RQ3 Can the bound $n^2/9$ be extended to $H$-free graphs where $H$ has chromatic number 4?
• RQ4 Does the extremal construction for $K_4$-free graphs suggest a general pattern for $K_r$-free graphs with $r > 4$?
• RQ5 Can the regularity lemma be used to derive tight bounds on edge deletions for $K_r$-free graphs?

Key findings

• Every $K_4$-free graph on $n$ vertices can be made bipartite by deleting at most $n^2/9$ edges.
• The complete 3-partite graph with parts of size $n/3$ is the only graph that requires exactly $n^2/9$ edge deletions.
• The bound $n^2/9$ is asymptotically tight and cannot be improved.
• For any fixed $H$ with $\chi(H) = 4$, any $H$-free graph on $n$ vertices can be made bipartite by deleting at most $(1+o(1))n^2/9$ edges.
• The result supports the conjecture of Chung and Graham on local density in $K_4$-free graphs.
• The proof technique using the regularity lemma and extremal graph structure provides a framework for extending results to higher $r$ in $K_r$-free graphs.