Cycle-Saturated Graphs with Minimum Number of Edges


  • Contract grant sponsors: Hungarian National Science Foundation OTKA; National Science Foundation (to Z. F.); Contract grant number: NFS DMS 09-01276 (to Z. F.).


A graph G is called H-saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph math formula contains some H. The minimum size of an n-vertex H-saturated graph is denoted by math formula. We prove

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holds for all math formula, where math formula is a cycle with length k. A graph G is H-semisaturated if math formula contains more copies of H than G does for math formula. Let math formula be the minimum size of an n-vertex H-semisaturated graph. We have

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We conjecture that our constructions are optimal for math formula. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013