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 $\mathsf{TS}{ω(G)-1}(G)$ from $\mathsf{TJ}{ω(G)}(G)$ and derive structural properties of $\mathsf{TJ}{ω(G)}(G)$ graphs, where $ω(G)$ 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.

💡 Research Summary

This paper presents a foundational study on the structural properties of reconfiguration graphs of cliques. Given a graph G and an integer k, the reconfiguration graph R_k(G) (for rules R ∈ {TS, TJ}) has vertices corresponding to all k-cliques of G, with an edge between two cliques if one can be transformed into the other via a single token slide (TS) or token jump (TJ) move.

The core technical contribution begins with Lemma 1, which characterizes the structure of complete subgraphs within TS_k(G). It proves that if vertices A1,…, An form a K_n in TS_k(G), then these k-cliques of G are either all obtained from a common (k-1)-clique ‘Int’ by adding n distinct vertices, or from a common (k+1)-clique ‘Uni’ by removing n distinct vertices. This lemma serves as the cornerstone for subsequent analysis.

Building on this, Theorem 2 establishes an exact formula relating the clique number of the reconfiguration graph to that of the original graph: ω(TS_k(G)) = max{k+1, ω(G) - k + 1} for k < ω(G). This result precisely quantifies how the complexity (clique size) of the reconfiguration space is determined by the original graph’s clique number and the size of the reconfigured cliques. The paper also bounds the chromatic number χ(TS_k(G)) by that of an appropriate Johnson graph J(ω(G), k).

The study then shifts to TJ reconfiguration graphs at the maximum clique size, i.e., TJ_ω(G)(G). The authors provide an algorithm to construct the graph TS_ω(G)-1(G) from TJ_ω(G)(G) and derive structural properties of these “maximum TJ graphs.”

Finally, the paper explores connections with planarity. A key result (Theorem) proves that if G is a planar graph, then TS_k(G) is also planar for any k. Leveraging this property and results about TS_k(G), the authors derive bounds on the number of small cliques in planar graphs. Notably, they prove that any planar graph with n vertices contains at most 3n - 8 triangles (3-cliques), which matches the classical bound for maximal planar graphs, providing a novel proof through the lens of reconfiguration graphs.

In summary, this work initiates the systematic graph-theoretic study of clique reconfiguration graphs, establishing fundamental relationships between their parameters (clique number, chromatic number, planarity) and those of the original graph, and offering new insights and proof techniques for classical planar graph bounds.