Abstract
A special cycle in a hypergraph is a cyclex 1 E 1 x 2 E 2 x 3 ...x n E n x 1 ofn distinct verticesx i andn distinct edgesE j (n≧3) whereE i ∩{x 1,x 2, ...,x n }={x i ,x i+1} (x n+1=x 1). In the equivalent (0, 1)-matrix formulation, a special cycle corresponds to a square submatrix which is the incidence matrix of a cycle of size at least 3. Hypergraphs with no special cycles have been called totally balanced by Lovász. Simple hypergraphs with no special cycles onm vertices can be shown to have at most ( m2 )+m+1 edges where the empty edge is allowed. Such hypergraphs with the maximum number of edges have a fascinating structure and are called solutions. The main result of this paper is an algorithm that shows that a simple hypergraph on at mostm vertices with no special cycles can be completed (by adding edges) to a solution.
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References
R. P. Anstee, Properties of (0, 1)-matrices with no triangles,J. Comb. Theory (A) 29 (1980), 186–198.
R. P. Anstee, Properties of (0, 1)-matrices without certain configurations,J. Comb. Theory (A),31 (1981), 256–269.
R. P. Anstee, Network flows approach for matrices with given row and column sums, Discrete Math.44 (1983), 125–138.
L. W. Beineke andR. E. Pippert, The number of labelledk-dimensional trees,J. Comb. Theory 6 (1969), 200–205.
C. Berge,Graphs and Hypergraphs, North-Holland, Amsterdam, 1973.
L. Lovász,Combinatorial Problems and Exercises, North-Holland, Amsterdam (1979).
H. J. Ryser, A fundamental matrix equation for finite sets,Proc. Amer. Math. Soc. 34 (1972), 332–336.
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Anstee, R. Hypergraphs with no special cycles. Combinatorica 3, 141–146 (1983). https://doi.org/10.1007/BF02579287
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DOI: https://doi.org/10.1007/BF02579287