Operadic structure on Hamiltonian paths and cycles

TL;DR

This paper introduces an operadic framework for analyzing Hamiltonian paths and cycles in graphs, using contractads and modules to encode their combinatorial structures and counting methods.

Contribution

It develops a novel operadic approach, called contractads, to model Hamiltonian paths and cycles, providing new algebraic tools for their enumeration.

Findings

Operad-like structure (contractad) for Hamiltonian paths.

Construction of a module for Hamiltonian cycles over the contractad.

Application of generating series for counting Hamiltonian structures.

Abstract

We study Hamiltonian paths and cycles in undirected graphs from an operadic viewpoint. We show that the graphical collection $\mathsf{Ham}$ encoding directed Hamiltonian paths in connected graphs admits an operad-like structure, called a contractad. Similarly, we construct the graphical collection of Hamiltonian cycles $\mathsf{CycHam}$ that forms a right module over the contractad $\mathsf{Ham}$. We use the machinery of contractad generating series for counting Hamiltonian paths/cycles for particular types of graphs.