Sparse Multipartite Graphs as Partition Universal for Graphs of Bounded-Degrees

TL;DR

This paper demonstrates that sparse random multipartite graphs are partition universal for bounded-degree graphs, extending previous results from complete and random graphs to multipartite structures using the sparse regularity lemma.

Contribution

It introduces the use of random multipartite graphs as partition universal graphs for bounded-degree graphs, generalizing earlier results from complete and Erdős–Rényi graphs.

Findings

Random multipartite graphs are partition universal for bounded-degree graphs.

The result holds for fixed maximum degree and large enough number of vertices.

The proof employs the sparse multipartite regularity lemma.

Abstract

For graphs $G$ and $H$, let $G\to (H,H)$ signify that any red/blue edge coloring of $G$ contains a monochromatic $H$ as a subgraph, and $\mathcal{H}(\Delta,n)=\{H:|V(H)|=n,\Delta(H)\le \Delta\}$. For fixed $\Delta$ and $n$, we say that $G$ is a partition universal graph for $\mathcal{H}(\Delta,n)$ if $G\to (H,H)$ for every $H\in\mathcal{H}(\Delta,n)$. In 1983, Chv\'atal, R\"odl, Szemer\'edi and Trotter proved that for any $\Delta\ge2$ there exists a constant $B$ such that, for any $n$, if $N\ge Bn$ then $K_N$ is partition universal for $\mathcal{H}(\Delta,n)$. Recently, Kohayakawa, R\"odl, Schacht and Szemer\'edi proved that the complete graph $K_N$ in above result can be replaced by sparse graphs. They obtained that for fixed $\Delta\ge2$, there exist constants $B$ and $C$ such that if $N\ge Bn$ and $p=C(\log N/N)^{1/\Delta}$, then {\bf a.a.s.} $G(N,p)$ is partition universal graph for $\mathcal{H}(\Delta,n)$, where $G(N,p)$ is the standard random graph on $N$ vertices with $\mathbb{P}(e)=p$ for each edge $e$. From some results of Bollob\'as and {\L}uczak, we know that {\bf a.a.s.} $\chi(G(N,p)) = \Theta((N/\log N)^{1-1/\Delta})$. In this paper, we shall show that the $G(N,p)$ in above result can be replaced by random multipartite graph. Let $K_{r}(N)$ be the complete $r$-partite graph with $N$ vertices in each part, and $G_r(N,p)$ the random spanning subgraph of $K_r(N)$, in which each edge appears with probability $p$. It is shown that for fixed $\Delta\ge2$ there exist constants $r, B$ and $C$ depending only on $\Delta$ such that if $N\ge Bn$ and $p=C(\log N/N)^{1/\Delta}$, then {\bf a.a.s.} $G_r(N,p)$ is partition universal graph for $\mathcal{H}(\Delta,n)$. The proof mainly uses the sparse multipartite regularity lemma.