Edge-transitive token graphs as covers

TL;DR

This work investigates edge-transitive token graphs by realizing selected families as covers of combined base graphs. It provides explicit constructions showing that, for even $n$, $F_2(K_n)$ is isomorphic to a cover $X^{(\alpha,\omega)}$ with a base on $n/2$ vertices and group $\mathbb{Z}_n$, and, for odd $n$, that $F_k(K_{1,n})$ with $k=(n+1)/2$ yields the doubled Johnson graphs as covers (with illustrative small cases) while $F_2(K_{1,n})$ can arise as a cover in other instances. The paper also states a conjecture: when $n$ divides $\binom{n+1}{2}$, $F_2(K_{1,n})$ is again a cover over a base graph with group $\mathbb{Z}_n$, highlighting a link between automorphism-free actions, covering theory, and quotient structures. Overall, it bridges covering-graph theory with edge-transitive token graphs, offering a framework to classify such graphs via combined base graphs and to explore quotient constructions driven by symmetries.

Abstract

This paper uses the theory of covering graphs to characterize some of the edge-transitive graphs which can arise as token graphs.

Figures (3)

• Figure 1: A combined based graph with cover isomorphic to $F_{2}(K_6)$
• Figure 2: The star $K_{1,5}$, its $3$-token graph, and the base graph whose cover is isomorphic to $F_3(K_{1,5})$
• Figure 3: The star $K_{1,5}$, its $2$-token graph, and the combined graph with cover isomorphic to $F_2(K_{1,5})$

Theorems & Definitions (6)

• Theorem 1.1: ZZ
• Theorem 3.1
• proof
• Example 3.2
• Conjecture 4.1
• Conjecture 5.1