On Generalized Token Graphs

TL;DR

This paper introduces generalized token graphs, called supertoken graphs, which extend traditional token graphs by varying token conditions and quantities, revealing diverse properties and applications.

Contribution

It defines supertoken graphs with flexible token conditions, broadening the scope of token graph applications and analyzing their fundamental properties.

Findings

Supertoken graphs include Cartesian powers of graphs as special cases.

Properties such as order, size, and connectivity are characterized for supertoken graphs.

Different token conditions lead to diverse graph families with unique features.

Abstract

The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex of $G$, we define a generalization of token graphs, which we call generalized token graphs or simply supertoken graphs, which have different applications. Depending on the above conditions, different families of graphs (such as the Cartesian $k$-th power of $G$ by itself) are obtained, and we present some of their properties, including order, size, and connectivity.