On the clique covering numbers of Johnson graphs

TL;DR

This paper studies the minimum number of cliques needed to cover vertices in Johnson graphs, providing exact values for certain parameters, bounds for general cases, and applications to coding and design theory.

Contribution

It introduces new identities for clique covering numbers in Johnson graphs and connects these to coding and combinatorial design theories, including optimal covers for specific cases.

Findings

Exact clique covering numbers for k ≤ 3 and k ≥ N - 3

Bounds on covering numbers for general Johnson graphs

Construction of optimal covers using constant-weight lexicodes

Abstract

We initiate a study of the vertex clique covering numbers of Johnson graphs $J(N, k)$, the smallest numbers of cliques necessary to cover the vertices of those graphs. We prove identities for the values of these numbers when $k \leq 3$, and $k \geq N - 3$, and using computational methods, we provide explicit values for a range of small graphs. By drawing on connections to coding theory and combinatorial design theory, we prove various bounds on the clique covering numbers for general Johnson graphs, and we show how constant-weight lexicodes can be utilized to create optimal covers of $J(2k, k)$ when $k$ is a small power of two.