Monochromatic bounded degree subgraph partitions

TL;DR

The paper proves that in any 2-edge-colored complete graph, vertices can be partitioned into a bounded number of monochromatic subgraphs with bounded maximum degree, with bounds depending on whether the subgraphs are bipartite.

Contribution

It establishes new bounds on monochromatic partitions into bounded degree subgraphs, improving previous results especially for bipartite graphs.

Findings

Partition size bounded by $2^{C ext{Delta} ext{log} ext{Delta}}$ for general graphs.

Improved bound $2^{C ext{Delta}}$ for bipartite graphs, which is optimal.

Results apply to any sequence of graphs with bounded maximum degree.

Abstract

Let ${\cal{F}}=\{F_1,F_2,\ldots\}$ be a sequence of graphs such that $F_n$ is a graph on $n$ vertices with maximum degree at most $\Delta$. We show that there exists an absolute constant $C$ such that the vertices of any 2-edge-colored complete graph can be partitioned into at most $2^{C\Delta \log{\Delta}}$ vertex disjoint monochromatic copies of graphs from ${\cal{F}}$. If each $F_n$ is bipartite, then we can improve this bound to $2^{C \Delta}$; this result is optimal up to the constant $C$.