Proper Hamiltonian Cycles in Edge-Colored Multigraphs

TL;DR

This paper establishes sufficient conditions for the existence of proper Hamiltonian cycles in edge-colored multigraphs, based on parameters like edge count and rainbow degree, advancing understanding in graph theory.

Contribution

It introduces new criteria linking multigraph parameters to the existence of proper Hamiltonian cycles, filling gaps in existing graph theory literature.

Findings

Identifies conditions on edge count for proper Hamiltonian cycles.

Relates rainbow degree to cycle existence.

Provides theoretical bounds for multigraph properties.

Abstract

A $c$-edge-colored multigraph has each edge colored with one of the $c$ available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree.