Extremal bipartite independence number and balanced coloring

TL;DR

This paper investigates extremal properties of bipartite graphs, establishing bounds on large balanced independent sets and partitions into independent sets, with results that improve previous theorems and are nearly optimal.

Contribution

It provides new algorithmic bounds on balanced independent sets and colorings in bipartite graphs, improving recent and existing theorems in the field.

Findings

Large bipartite graphs have balanced independent sets of size proportional to (log D)/D times n.

Balanced bipartite graphs can be partitioned into approximately (Δ/log Δ) independent sets.

Results are algorithmic and nearly optimal, highlighting the 'algorithmic barrier' phenomenon.

Abstract

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $D$ and with $n$ vertices on each side has a balanced independent set containing $(1-\epsilon) \frac{\log D}{D} n$ vertices from each side for small $\epsilon > 0$. Secondly, we prove that the vertex set of every sufficiently large balanced bipartite graph with maximum degree at most $\Delta$ can be partitioned into $(1+\epsilon)\frac{\Delta}{\log \Delta}$ balanced independent sets. Both of these results are algorithmic and best possible up to a factor of 2, which might be hard to improve as evidenced by the phenomenon known as `algorithmic barrier' in the literature. The first result improves a recent theorem of Axenovich, Sereni, Snyder, and Weber in a slightly more general setting. The second result improves a theorem of Feige and Kogan about coloring balanced bipartite graphs.