Distance-preserving subgraphs of Johnson graphs

TL;DR

This paper characterizes distance-preserving subgraphs of Johnson graphs, providing an explicit description of the wallspace that defines their isometric embedding, extending previous work on hypercube subgraphs.

Contribution

It offers a new characterization of distance-preserving subgraphs of Johnson graphs, similar to Djoković's work on hypercubes, with an explicit description of the embedding's wallspace.

Findings

Provides a characterization of isometric subgraphs of Johnson graphs.

Extends Djoković's hypercube subgraph characterization to Johnson graphs.

Offers an explicit description of the wallspace for embeddings.

Abstract

We give a characterization of distance--preserving subgraphs of Johnson graphs, i.e. of graphs which are isometrically embeddable into Johnson graphs (the Johnson graph $J(m,\Lambda)$ has the subsets of cardinality $m$ of a set $\Lambda$ as the vertex--set and two such sets $A,B$ are adjacent iff $|A\triangle B|=2$). Our characterization is similar to the characterization of D. \v{Z}. Djokovi\'c (J. Combin. Th. Ser. B 14 (1973), 263--267) of distance--preserving subgraphs of hypercubes and provides an explicit description of the wallspace (split system) defining the embedding.