Token Graphs

TL;DR

This paper introduces token graphs, a new way to model configurations of indistinguishable tokens on a graph, and explores their fundamental properties such as connectivity, diameter, and Hamiltonian paths.

Contribution

It defines token graphs based on symmetric differences and investigates their structural properties, a novel approach in graph theory.

Findings

Token graphs' connectivity and diameter are characterized.

Conditions for Hamiltonian paths in token graphs are identified.

Properties like cliques and chromatic number are analyzed.

Abstract

For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.