Equidistant dimension of Johnson and Kneser graphs

TL;DR

The paper studies the equidistant dimension $eqdim(G)$ for Johnson graphs $J_{n,k}$ and Kneser graphs $K_{n,k}$, introducing the distance-equalizer framework. It develops structural properties of distance-equalizer sets and constructs explicit examples to obtain exact values and tight bounds. Key findings include $eqdim(J_{n,2})=3$ for $n\ge6$, $eqdim(K_{n,2})=3$, and $eqdim(J_{2k,k})=\frac{1}{2}\binom{2k}{k}$ for odd $k$, as well as $eqdim(J_{n,3})\le n-2$ for $n\ge9$. These results advance understanding of the equidistant dimension as a graph invariant and motivate further study of this parameter for other graph families and algorithmic approaches.

Abstract

In this paper the recently introduced concept of equidistant dimension $eqdim(G)$ of graph $G$ is considered. Useful property of distance-equalizer set of arbitrary graph $G$ has been established. For Johnson graphs $J_{n,2}$ and Kneser graphs $K_{n,2}$ exact values for $eqdim(J_{n,2})$ and $eqdim(K_{n,2})$ have been derived, while for Johnson graphs $J_{n,3}$ it is proved that $eqdim(J_{n,3}) \le n-2$. Finally, exact value of $eqdim(J_{2k,k})$ for odd $k$ has been presented.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Corollary 1
• Theorem 1
• Corollary 2
• Theorem 2
• Corollary 3
• Proposition 1