Regularity and Planarity of Token Graphs
Walter Carballosa, Ruy Fabila-Monroy, Jes\'us Lea\~nos, Luis Manuel, Rivera

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 be a graph of order and let be an integer. The -token graph of is the graph whose vertices are all the -subsets of , two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in . In this paper we characterize precisely, for each value of , which graphs have a regular -token graph and which connected graphs have a planar -token graph.
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