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Keywords:

  • independence;
  • matchings;
  • hypergraphs;
  • rank;
  • ACM Classification;
  • 05C69

Abstract

In this paper, we study lower bounds on the size of a maximum independent set and a maximum matching in a hypergraph of rank three. The rank in a hypergraph is the size of a maximum edge in the hypergraph. A hypergraph is simple if no two edges contain exactly the same vertices. Let H be a hypergraph and let inline image and inline image be the size of a maximum independent set and a maximum matching, respectively, in H, where a set of vertices in H is independent (also called strongly independent in the literature) if no two vertices in the set belong to a common edge. Let H be a hypergraph of rank at most three and maximum degree at most three. We show that inline image with equality if and only if H is the Fano plane. In fact, we show that if H is connected and different from the Fano plane, then inline image and we characterize the hypergraphs achieving equality in this bound. Using this result, we show that that if H is a simple connected 3-uniform hypergraph of order at least 8 and with maximum degree at most three, then inline image and there is a connected 3-uniform hypergraph H of order 19 achieving this lower bound. Finally, we show that if H is a connected hypergraph of rank at most three that is not a complete hypergraph on inline image vertices, where inline image denotes the maximum degree in H, then inline image and this bound is asymptotically best possible. © 2012 Wiley Periodicals, Inc. J. Graph Theory