Regularity and planarity of token graphs

Abstract

Let G = (V, E) be a graph of order n and let 1 ≤ 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.

The 4-girth-thickness of the complete graph

“…In [8], the authors remark that the 2-token graph F 2 (P n ) of the path graph P n of n vertices is planar of girth at least 4 for every n. Now, we prove that an edge-partition of a graph G induces an edge partition of F k (G). Lemma 2.1.…”

Well-covered token graphs

“…Various properties of token graphs have recently been studied. For example, in [6], Carballosa, Fabila-Monroy, Leaños, and Rivera characterize when the token graphs are regular, as well as when a token graph is planar. In [16], Leaños and Trujillo-Negrete prove a conjecture about the connectivity of token graphs.…”

Hamiltonicity of the Complete Double Vertex Graph of some Join Graphs

“…The properties of token graphs have been studied since 1991 by various authors and with different names, see, e.g., [1,2,3,5,9] and, in recent years, the study of its combinatorial properties and its connection with problems in other areas such as Coding Theory and Physics has increased, see, e.g., [4,6,8,10,11,12,14,15]. However, there are only few papers about complete double vertex graphs, which were implicitly introduced by Chartrand et al [7].…”