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The Binomial k ‐Clique
Author(s) -
Narayaithya Sai,
Sane Sharad
Publication year - 2013
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21333
Subject(s) - mathematics , combinatorics , clique , intersection (aeronautics) , discrete mathematics , binomial (polynomial) , statistics , engineering , aerospace engineering
A finite collection C of k ‐sets, where k ≥ 2 , is called a k ‐clique if every two k ‐sets (called lines) in C have a nonempty intersection and a k ‐clique is a called a maximal k ‐clique if | C | < ∞ and C is maximal with respect to this property. That is, every two lines in C have a nonempty intersection and there does not exist A such that | A | ≤ k , A ∉ C and A ∩ X ≠ ∅ for all X ∈ C . An elementary example of a maximal k ‐clique is furnished by the family of all the k ‐subsets of a ( 2 k − 1 ) ‐set. This k ‐clique will be called the binomial k ‐clique. This paper is intended to give some combinatorial characterizations of the binomial k ‐clique as a maximal k ‐clique. The techniques developed are then used to provide a large number of examples of mutually nonisomorphic maximal k ‐cliques for a fixed value of k .