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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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
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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 ≤ 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.
AbstractList 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.
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(2k, k) when k is a small power of two.
Author Jørgensen, Søren Fuglede
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Cites_doi 10.1137/090765596
10.1007/BF01929486
10.1109/TIT.1962.1057714
10.1201/9781420010541
10.1109/TIT.2010.2050922
10.1109/18.59932
10.3934/amc.2011.5.417
10.1109/18.887851
10.1017/CBO9781316414958
10.1002/jcd.10022
10.1109/TIT.1986.1057187
10.1080/00029890.1959.11989408
10.1109/TIT.1980.1056141
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The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.
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Constant-weight codes
Clique covers
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Snippet 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...
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SubjectTerms 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
Subtitle On the clique covering numbers of Johnson graphs
Title On the clique covering numbers of Johnson graphs
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