A note on equitable Hamiltonian cycles

Tim Ophelders,Roel Lambers,Frits C.R Spieksma,Tjark Vredeveld

doi:10.1016/j.dam.2020.08.023

Abstract

Given a complete graph with an even number of vertices, and with each edge colored with one of two colors (say red or blue), an equitable Hamiltonian cycle is a Hamiltonian cycle that can be decomposed into two perfect matchings such that both perfect matchings have the same number of red edges. We show that, for any coloring of the edges, in any complete graph on at least 6 vertices, an equitable Hamiltonian cycle exists.

Highlights

Let G = (V, E) be a simple, complete graph on n ≡ |V | vertices, with n even
In Appendix A we argue that such an equitable Hamiltonian cycle exists
The running time of the algorithm can be improved by the following observation, which is shown in Lemma 3.2: any equitable Hamiltonian cycle on at least 12 vertices has at least two mono-chromatic pairs that have at least two edges in between

Summary

Introduction

Any Hamiltonian cycle C present in the colored graph G, can be decomposed into two perfect matchings ME (C ) and MO(C ). We investigate whether colored graphs are nice, and we address algorithms that identify equitable Kinable et al [11] are interested in finding solutions to the Traveling Salesman Problem where, instead of minimizing total cost, the absolute value of the difference between the costs of the two perfect matchings making up the tour is minimized; they call this variant the Equitable TSP. We view the problem considered here as related to Ramsey theory: we prove that colored graphs of a certain size necessarily contain equitable Hamiltonian cycles.

General outline

Merging two equitable cycles

The running time of Algorithm 1

About 2-OPT

Concluding remarks