Decomposition of bounded degree graphs into $C_4$-free subgraphs

TL;DR

The paper proves that graphs with bounded maximum degree can be partitioned into a near-optimal number of $C_4$-free subgraphs using an iterated random coloring method, advancing understanding of graph decompositions.

Contribution

It establishes a tight bound on the number of $C_4$-free subgraphs needed to partition graphs with bounded degree, employing a novel iterative random coloring technique.

Findings

Partition size is $O(\sqrt{\Delta})$ for graphs with maximum degree $\Delta$.

The bound is proven to be sharp up to a constant factor.

The proof introduces an iterated random coloring approach.

Abstract

We prove that every graph with maximum degree $\Delta$ admits a partition of its edges into $O(\sqrt{\Delta})$ parts (as $\Delta\to\infty$) none of which contains $C_4$ as a subgraph. This bound is sharp up to a constant factor. Our proof uses an iterated random colouring procedure.