Triangle-different Hamiltonian paths

TL;DR

This paper determines the maximum number of Hamiltonian paths in a graph that are pairwise distinguishable by a triangle subgraph, linking it to the count of balanced bipartitions.

Contribution

It proves that the maximum number of pairwise triangle-different Hamiltonian paths equals the number of balanced bipartitions, answering a question posed by K"orner, Messuti, and Simonyi.

Findings

Maximum number of pairwise triangle-different Hamiltonian paths equals the number of balanced bipartitions.

Established a direct link between Hamiltonian path differences and bipartition counts.

Resolved an open question in graph theory regarding Hamiltonian path distinguishability.

Abstract

Let $G$ be a fixed graph. Two paths of length $n-1$ on $n$ vertices (Hamiltonian paths) are $G$-different if there is a subgraph isomorphic to $G$ in their union. In this paper we prove that the maximal number of pairwise triangle-different Hamiltonian paths is equal to the number of balanced bipartitions of the ground set, answering a question of K\"orner, Messuti and Simonyi.