A Note on Reconfiguration Graphs of Cliques

TL;DR

This paper explores the structural properties of reconfiguration graphs of cliques in a graph, focusing on rules like Token Sliding and Token Jumping, and establishes bounds and algorithms related to clique reconfigurations.

Contribution

It introduces new formulas, bounds, and algorithms for analyzing reconfiguration graphs of cliques, especially under Token Sliding and Token Jumping rules, and relates these to classical graph properties.

Findings

Established a formula relating clique numbers of G and its reconfiguration graphs.

Bound the chromatic number of reconfiguration graphs using Johnson graphs.

Proved planarity preservation and bounds on the number of small cliques in planar graphs.

Abstract

In a reconfiguration setting, each clique of a graph $G$ is viewed as a set of tokens placed on vertices of $G$ such that no vertex has more than one token and any two tokens are adjacent. Three well-known reconfiguration rules have been studied in the literature: Token Jumping ($\mathsf{TJ}$), Token Sliding ($\mathsf{TS}$), and Token Addition/Removal ($\mathsf{TAR}$). Given a graph $G$ and a reconfiguration rule $\mathsf{R} \in \{\mathsf{TS}, \mathsf{TJ}, \mathsf{TAR}\}$, a reconfiguration graph of $k$-cliques of $G$, denoted by $\mathsf{R}_k(G)$, is the graph whose vertices are cliques of $G$ of size $k$ and two vertices are adjacent if one can be obtained from the other by applying $\mathsf{R}$ exactly once. In this paper, we initiate the study of structural properties of reconfiguration graphs of cliques, proving several interesting results primarily under $\mathsf{TS}$ and $\mathsf{TJ}$ rules. In particular, we establish a formula relating the clique number of $G$ and that of $\mathsf{TS}_k(G)$, and bound the chromatic number of $\mathsf{TS}_k(G)$ via that of an appropriate Johnson graph. Additionally, we present an algorithm to construct $\mathsf{TS}_{\omega(G)-1}(G)$ from $\mathsf{TJ}_{\omega(G)}(G)$ and derive structural properties of $\mathsf{TJ}_{\omega(G)}(G)$ graphs, where $\omega(G)$ denotes the clique number of $G$. Finally, we show that $\mathsf{TS}_k(G)$ is planar whenever $G$ is planar and establish bounds on the number of $3$- and $4$-cliques based on results concerning $\mathsf{TS}_k(G)$ graphs. In particular, we prove that any planar graph $G$ with $n$ vertices can contain at most $3n - 8$ triangles, which aligns with the classical bound on maximal planar graphs.