A Note on Reconfiguration Graphs of Cliques

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.