Regularity and Planarity of Token Graphs

TL;DR

This paper characterizes when the $k$-token graph of a graph is regular or planar, providing precise conditions for these properties depending on the original graph and the value of $k$.

Contribution

It offers a complete characterization of regular and planar $k$-token graphs for all values of $k$, advancing understanding of token graph properties.

Findings

Identifies graphs with regular $k$-token graphs for each $k$.

Determines which connected graphs have planar $k$-token graphs.

Provides exact conditions for regularity and planarity of token graphs.

Abstract

Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper we characterize precisely, for each value of $k$, which graphs have a regular $k$-token graph and which connected graphs have a planar $k$-token graph.