Bounded colorings of multipartite graphs and hypergraphs

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 Bollobás and Erdős and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruciński [9]. 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 ∆. Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes ∆ ≫ m and ∆ ≪ m Our main tool is the framework of Lu and Székely for the space of random bijections, which we extend to product spaces.

Rainbow spanning subgraphs in bounded edge–colourings of graphs with large minimum degree

“…Other related results include the existence of almost spanning rainbow trees in K n [7], the existence of spanning rainbow multipartite graphs with bounded degrees in multipartite complete graphs K n,n...,n [8] as well as the existence of rainbow subhypergraphs in complete hypergraphs [5,8]. All these results hold provided that the edge-coloring of the host graph is k-bounded for an appropriate value of k.…”

“…Sudakov and Volec gave a first step on this direction by proving the following (see the concluding remarks in [13], or Theorem 1.7 in [8] for a formal statement).…”

A rainbow blow‐up lemma

“…Using these tools, they show that a random injection V ( H )→ V ( K n ) yields with positive probability a rainbow copy of H . Kamčev, Sudakov, and Volec recently extended this result to the setting where G is complete multipartite, and Sudakov and Volec considered the case when the number of cherries in H is restricted (instead of the maximum degree).…”

Long directed rainbow cycles and rainbow spanning trees

“…A partial transversal is called cycle-free is indeed optimal. See [11,23] for additional work in this area when the host graph Kn is replaced by a different graph.…”