Colored graphs without colorful cycles

TL;DR

This paper characterizes colored graphs without colorful cycles, explores the algebraic structure of omitted cycle lengths, and provides structural descriptions of exact Gallai graphs and their monochromatic components.

Contribution

It introduces a new algebraic framework for omitted cycle lengths and characterizes exact Gallai graphs through iterative construction and homomorphism duality.

Findings

Colored graphs without colorful cycles are exactly Gallai graphs.

Omitted cycle lengths form a monoid isomorphic to a submonoid of natural numbers.

Exact Gallai graphs can be built from three simple base graphs.

Abstract

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation $m\circ n\equiv m+n-2$, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having $2$, $4$, and $5$ vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality.