Bounded colorings of multipartite graphs and hypergraphs

TL;DR

This paper extends classical results on properly colored and rainbow Hamilton cycles from complete graphs to hypergraphs and multipartite graphs, providing optimal conditions for the existence of such structures under various coloring restrictions.

Contribution

It generalizes previous results to hypergraphs and multipartite graphs, establishing near-optimal conditions for the existence of properly colored and rainbow subgraphs with maximum degree 4, revealing a phase transition in growth regimes.

Findings

Established sufficient conditions for hypergraphs and multipartite graphs

Identified a phase transition in growth regimes 4 and 4

Extended the framework of Lu and Sze9kely to product spaces

Abstract

Let $c$ be an edge-coloring of the complete $n$-vertex graph $K_n$. The problem of finding properly colored and rainbow Hamilton cycles in $c$ was initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruci\'nski. We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph $G$. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal sufficient conditions for a coloring $c$ of the complete balanced $m$-partite graph to contain a properly colored or rainbow copy of a given graph $G$ with maximum degree $\Delta$. Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes $\Delta \gg m$ and $\Delta \ll m$ Our main tool is the framework of Lu and Sz\'ekely for the space of random bijections, which we extend to product spaces.