A Note on Reconfiguration Graphs of Cliques

Quan N. Lam; Huu An Phan; Duc A. Hoang

Paper

A Note on Reconfiguration Graphs of Cliques

Abstract

In a reconfiguration setting, each clique of a graph

is viewed as a set of tokens placed on vertices of

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 (

), Token Sliding (

), and Token Addition/Removal (

). Given a graph

and a reconfiguration rule

, a reconfiguration graph of

-cliques of

, denoted by

, is the graph whose vertices are cliques of

of size

and two vertices are adjacent if one can be obtained from the other by applying

exactly once. In this paper, we initiate the study of structural properties of reconfiguration graphs of cliques, proving several interesting results primarily under

and

rules. In particular, we establish a formula relating the clique number of

and that of

, and bound the chromatic number of

via that of an appropriate Johnson graph. Additionally, we present an algorithm to construct

from

and derive structural properties of

graphs, where

denotes the clique number of

. Finally, we show that

is planar whenever

is planar and establish bounds on the number of

- and

-cliques based on results concerning

graphs. In particular, we prove that any planar graph

with

vertices can contain at most

triangles, which aligns with the classical bound on maximal planar graphs.