Proper Hamiltonian Paths in Edge-Coloured Multigraphs

TL;DR

This paper establishes sufficient conditions involving edges, colours, rainbow degree, and connectivity in edge-coloured multigraphs to guarantee the existence of proper Hamiltonian paths, advancing understanding in graph theory.

Contribution

It provides new sufficient conditions for the existence of proper Hamiltonian paths in edge-coloured multigraphs based on multiple graph parameters.

Findings

Identifies conditions involving edges, colours, and connectivity for Hamiltonian paths.

Extends previous results by incorporating rainbow degree.

Offers theoretical bounds for Hamiltonian path existence.

Abstract

Given a $c$-edge-coloured multigraph, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we establish sufficient conditions for an edge-coloured multigraph to guarantee the existence of a proper Hamiltonian path, involving various parameters as the number of edges, the number of colours, the rainbow degree and the connectivity.