[Paper Review] Dense induced bipartite subgraphs in triangle-free graphs

Matthew Kwan, Shoham Letzter+2 more|arXiv (Cornell University)|Oct 29, 2018

Limits and Structures in Graph Theory19 references4 citations

TL;DR

This paper establishes that every $H$-free graph with minimum degree $d$ contains an induced bipartite subgraph of minimum degree at least $c_H \log d / \log \log d$, nearly confirming a conjecture by Esperet, Kang, and Thomassé. The result is tight up to logarithmic factors and extends to dense triangle-free graphs, where the bound improves to $\Theta(\log d)$, resolving a long-standing problem in extremal graph theory.

ABSTRACT

The problem of finding dense induced bipartite subgraphs in $H$-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any $K_t$-free graph with minimum degree at least $d$ contains an induced bipartite subgraph of minimum degree at least $c_t \log d/\log \log d$, confirming (asymptotically) several conjectures by Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdős, Janson, Łuczak and Spencer.

Motivation & Objective

• To resolve a longstanding conjecture by Erdős, Faudree, Pach, and Spencer on induced bipartite subgraphs in triangle-free graphs.
• To establish tight lower bounds on the minimum degree of induced bipartite subgraphs in $H$-free graphs with minimum degree $d$.
• To address the phase transition behavior of induced bipartite subgraph minimum degree in dense triangle-free graphs.
• To improve upon prior results on the Max Cut problem and induced bipartite subgraphs in restricted graph classes.
• To explore connections between induced bipartite subgraphs and fractional chromatic number in $d$-degenerate triangle-free graphs.

Proposed method

• Prove the main result for triangle-free graphs using probabilistic and extremal combinatorial techniques.
• Reduce the general $H$-free case to the triangle-free case via a reduction argument based on graph blowups and induced subgraph properties.
• Use random graph constructions to show the optimality of the $\log d / \log \log d$ bound up to logarithmic factors.
• Analyze the function $g(n,d)$, the maximum minimum degree of induced bipartite subgraphs in $n$-vertex, $d$-minimum-degree triangle-free graphs, identifying a phase transition at $d = \sqrt{n}$.
• Construct extremal graphs via blowups of specific triangle-free graphs to establish upper bounds on $g(n,d)$.
• Leverage connections to fractional coloring and chromatic number to explore alternative proof strategies and conjecture tightness of bounds.

Experimental results

Research questions

• RQ1 What is the largest minimum degree of an induced bipartite subgraph in an $H$-free graph with minimum degree $d$?
• RQ2 Can the bound $\Omega(\log d / \log \log d)$ for induced bipartite subgraphs in $H$-free graphs be improved to $\Omega(\log d)$?
• RQ3 How does the maximum minimum degree of induced bipartite subgraphs in triangle-free graphs behave as a function of $n$ and $d$?
• RQ4 Is the fractional chromatic number of $d$-degenerate triangle-free graphs bounded by $O(d / \log d)$, and what does this imply for induced bipartite subgraphs?
• RQ5 What is the precise threshold behavior of induced bipartite subgraph minimum degree in triangle-free graphs, particularly around $d = \sqrt{n}$?

Key findings

• For every fixed graph $H$, every $H$-free graph with minimum degree $d$ contains an induced bipartite subgraph of minimum degree at least $c_H \log d / \log \log d$, with $c_H \geq \varepsilon^{|H|}$ for some $\varepsilon > 0$.
• The bound $\Omega(\log d / \log \log d)$ is nearly optimal, as random graph constructions show that no $\Omega(\log d)$ bound is possible in general.
• For triangle-free graphs with $d \leq \sqrt{n}$, the maximum minimum degree of an induced bipartite subgraph is $\Theta(\log d)$, confirming a tight bound.
• For triangle-free graphs with $\sqrt{n} < d \leq n/2$, the maximum minimum degree of an induced bipartite subgraph is $\Theta(d^2 \log d / n)$, showing a significant drop in density.
• When $d \geq n^{2/3}$, the maximum minimum degree of an induced bipartite subgraph is $O(d^2 / n)$, and this bound is tight up to logarithmic factors.
• The upper bound for $g(n,d)$ in the dense regime ($d \geq n^{2/3}$) is achieved via blowup constructions of specific triangle-free graphs, demonstrating the tightness of the result.