Restricted subgraphs of edge-colored graphs and applications

TL;DR

This survey addresses the problem of identifying rainbow and other edge-constrained subgraphs within edge-colored graphs, motivated by connections to Latin squares and design theory. It organizes results across cycles, matchings, trees, and decompositions, highlighting deep links to coding theory via even covers, and to additive combinatorics through rainbow substructures. The authors synthesize advances in bounds for rainbow cycles, large rainbow matchings, and asymptotic decompositions (notably Ringel-type results) using techniques such as probabilistic nibble, expansion methods, homomorphism counts, and absorption. The work emphasizes the interdisciplinary impact of rainbow subgraph questions, with applications spanning graph decomposition, coding theory, theoretical computer science, and harmonic analysis, and provides a cohesive framework for future developments.

Abstract

A properly edge-colored graph is a graph with a coloring of its edges such that no vertex is incident to two or more edges of the same color. A subgraph is called rainbow if all its edges have different colors. The problem of finding rainbow subgraphs or other restricted structures in edge-colored graphs has a long history, dating back to Euler's work on Latin squares. It has also proven to be a powerful method for studying several well-known questions in other areas. In this survey, we will provide a brief introduction to this topic, discuss several results in this area, and demonstrate their applications to problems in graph decomposition, additive combinatorics, theoretical computer science, and coding theory.

Figures (5)

• Figure 1: Correspondence between a Latin square and a properly edge-colored complete bipartite graph, on the left, and between a transversal and rainbow perfect matching, on the right.
• Figure 2: Cayley sum graph of the group $\mathbb{F}_2^3$ with respect to the standard basis and its canonical edge coloring.
• Figure 3: An illustration of the Kikuchi graph defined above with an edge $E = \{1,2,3,4\}$ of ${\cal H}$ and an edge $S \xleftrightarrow{} T$ of $G$ colored with $E$.
• Figure 4: The addition table of the cyclic group $\mathbb{Z}_4$, for which the corresponding Latin square does not contain a full transversal.
• Figure 5: The ND-coloring of $K_9$ and a rainbow copy of a tree $T$ with four edges. The color of each edge corresponds to its Euclidean length. By taking cyclic shifts of this tree around the centre of the picture we obtain $9$ disjoint copies of the tree decomposing $K_9$ (and thus a proof of Ringel's Conjecture for this particular tree). To see that this gives 9 disjoint trees, notice that each edge must be shifted to the other edges of the same color (since shifts are isometries).

Theorems & Definitions (23)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Conjecture 2.5
• Theorem 2.6
• Theorem 2.7
• Conjecture 2.8
• Theorem 2.9
• Theorem 2.10
• Definition 2.11