A Rainbow Dirac's Theorem

TL;DR

This paper extends Dirac's theorem by proving that in certain edge-colored Dirac graphs with limited colors, rainbow Hamilton cycles are guaranteed to exist, broadening understanding of Hamiltonicity under coloring constraints.

Contribution

It introduces a new result showing rainbow Hamilton cycles in bounded-color Dirac graphs, strengthening classical Hamiltonicity theorems with coloring considerations.

Findings

Rainbow Hamilton cycles exist in bounded-color Dirac graphs for small enough color bounds.

The result generalizes classical Hamiltonicity to edge-colored graphs with coloring restrictions.

Provides a new perspective on Hamilton cycles in colored graph settings.

Abstract

A famous theorem of Dirac states that any graph on $n$ vertices with minimum degree at least $n/2$ has a Hamilton cycle. Such graphs are called Dirac graphs. Strengthening this result, we show the existence of rainbow Hamilton cycles in $\mu n$-bounded colourings of Dirac graphs for sufficiently small $\mu >0$.