How to construct the symmetric cycle of length 5 using Haj\'os construction with an adapted Rank Genetic Algorithm

TL;DR

This paper presents an innovative genetic algorithm approach with specialized operators to efficiently construct the symmetric cycle of length 5 from a complete symmetric digraph using Hajós constructions.

Contribution

The authors adapt a rank-based genetic algorithm with novel graph-theoretic operators to solve a complex Hajós construction problem for digraphs.

Findings

Successfully constructed the symmetric 5-cycle using only 16 Hajós operations.

Introduced new recombination and mutation operators based on graph theory.

Demonstrated the effectiveness of genetic algorithms in solving complex graph construction problems.

Abstract

In 2020 Bang-Jensen et. al. generalized the Haj\'os join of two graphs to the class of digraphs and generalized several results for vertex colorings in digraphs. Although, as a consequence of these results, a digraph can be obtained by Haj\'os constructions (directed Haj\'os join and identifying non-adjacent vertices), determining the Haj\'os constructions to obtain the digraph is a complex problem. In particular, Bang-Jensen et al. posed the problem of determining the Haj\'os operations to construct the symmetric 5-cycle from the complete symmetric digraph of order 3 using only Haj\'os constructions. We successfully adapted a rank-based genetic algorithm to solve this problem by the introduction of innovative recombination and mutation operators from graph theory. The Haj\'os Join became the recombination operator and the identification of independent vertices became the mutation operator. In this way, we were able to obtain a sequence of only 16 Haj\'os operations to construct the symmetric cycle of order 5.