Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique |
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Authors: | Myung S Chung Douglas B West |
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Institution: | (1) NCBI/National Institute of Health, 20894 Maryland, Bethesda, USA;(2) University of Illinois, 61801 Illinois, Urbana, USA |
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Abstract: | Thep-intersection graph of a collection of finite sets {S
i
}
i=1
n
is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S
i
S
j
|p. Thep-intersection number of a graphG, herein denoted
p
(G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK
n,n
andp2, then
p
(K
n, n
)(n
2+(2p–1)n)/p. Whenp=2, equality holds if and only ifK
n
has anorthogonal double covering, which is a collection ofn subgraphs ofK
n
, each withn–1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K
n
has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.Research supported in part by ONR Grant N00014-5K0570. |
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Keywords: | 05 C 35 05 C 70 90 C 90 |
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