Rainbow Graphs and Switching Classes

TL;DR

This paper explores the properties of rainbow graphs, a special class of vertex-colored graphs, and establishes a bijection between rainbow graphs on 2n vertices and switching classes of graphs on n vertices, revealing a deep combinatorial connection.

Contribution

It introduces a novel bijection linking rainbow graphs and switching classes, advancing understanding of their structural relationship.

Findings

Established a bijection between n-rainbow graphs on 2n vertices and switching classes on n vertices.

Provided insights into the structural properties of rainbow graphs.

Connected rainbow graphs to graph switching classes, enriching combinatorial graph theory.

Abstract

A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection between the isomorphism classes of n-rainbow graphs on 2n vertices and the switching classes of graphs on n vertices.