Well-covered Token Graphs

TL;DR

This paper investigates the conditions under which token graphs derived from a base graph are well-covered, providing classifications for bipartite graphs and bounds for general graphs, with specific focus on the case when k=2.

Contribution

It classifies when token graphs of bipartite graphs are well-covered and establishes girth constraints for general graphs with well-covered token graphs.

Findings

Classified when T_k(G) is well-covered for bipartite graphs

Showed girth of G is at most four if T_2(G) is well-covered

Provided bounds on independence number of T_k(G)

Abstract

The $k$-token graph $T_k(G)$ is the graph whose vertices are the $k$-subsets of vertices of a graph $G$, with two vertices of $T_k(G)$ adjacent if their symmetric difference is an edge of $G$. We explore when $T_k(G)$ is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs $G$, we classify when $T_k(G)$ is well-covered. For an arbitrary graph $G$, we show that if $T_2(G)$ is well-covered, then the girth of $G$ is at most four. We include upper and lower bounds on the independence number of $T_k(G)$, and provide some families of well-covered token graphs.