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Towards Ryser's Conjecture: Bounds on the Cardinality of Partitioned Intersecting Hypergraphs
Anna E. Dodson
Dartmouth TR2020-902

Abstract: This work is motivated by the open conjecture concerning the size of a minimum vertex cover in a partitioned hypergraph. In an r-uniform r-partite hypergraph, the size of the minimum vertex cover C is conjectured to be related to the size of its maximum matching M by the relation (|C|<= (r-1)|M|). In fact it is not known whether this conjecture holds when |M| = 1. We consider r-partite hypergraphs with maximal matching size |M| = 1, and pose a novel algorithmic approach to finding a vertex cover of size (r - 1) in this case. We define a reactive hypergraph to be a back-and-forth algorithm for a hypergraph which chooses new edges in response to a choice of vertex cover, and prove that this algorithm terminates for all hypergraphs of orders r = 3 and 4. We introduce the idea of optimizing the size of the reactive hypergraph and find that the reactive hypergraph terminates for r = 5...20. We then consider the case where the intersection of any two edges is exactly 1. We prove bounds on the size of this 1-intersecting hypergraph and relate the 1-intersecting hypergraph maximization problem to mutually orthogonal Latin squares. We propose a generative algorithm for 1-intersecting hypergraphs of maximal size for prime powers r-1 = pd under the constraint pd+1 is also a prime power of the same form, and therefore pose a new generating algorithm for MOLS based upon intersecting hypergraphs. We prove this algorithm generates a valid set of mutually orthogonal Latin squares and prove the construction guarantees certain symmetric properties. We conclude that a conjecture by Lovasz, that the inequality in Ryser's Conjecture cannot be improved when (r-1) is a prime power, is correct for the 1-intersecting hypergraph of prime power orders.

Note: Senior Honors Thesis. Advisor: Deeparnab Chakrabarty.

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   Anna E. Dodson, "Towards Ryser's Conjecture: Bounds on the Cardinality of Partitioned Intersecting Hypergraphs." Dartmouth Computer Science Technical Report TR2020-902, June 2020.

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