On the clique covering numbers of Johnson graphs On the clique covering numbers of Johnson graphs

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 ≤ 3 , and k ≥ N - 3 , and using computational methods, we provide expl...

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Bibliographic Details
Published in:Designs, codes, and cryptography Vol. 93; no. 9; pp. 3689 - 3705
Main Author: Jørgensen, Søren Fuglede
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2025
Springer Nature B.V
Subjects:
Algorithms
Apexes
Codes
Coding and Information Theory
Coding theory
Combinatorial analysis
Computer Science
Cryptology
Discrete Mathematics in Computer Science
Graphs
Integer programming
Linear programming
Optimization
Johnson graphs
Constant-weight codes
Clique covers
ISSN:0925-1022, 1573-7586
Online Access:Get full text
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Summary: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 ≤ 3 , and k ≥ 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 (2 k ,  k ) when k is a small power of two.
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ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-025-01663-3