Cycle-Saturated Graphs with Minimum Number of Edges

Abstract

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 G+e contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by sat(n,H). We prove sat(n,Ck)=n+n/k+O((n/k2)+k2)holds for all n≥k≥3, where Ck is a cycle with length k. A graph G is H‐semisaturated if G+e contains more copies of H than G does for ∀e∈E(G¯). Let ssat (n,H) be the minimum size of an n‐vertex H‐semisaturated graph. We have ssat(n,Ck)=n+n/(2k)+O((n/k2)+k).We conjecture that our constructions are optimal for n>n0(k). © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013

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“…Since each y ∈ S 2 is adjacent to two vertices in f (y) ∩ U 1 3 , and the vertices in S 1 satisfy (7), it follows 2|S 2 | ≤ 2|U 1 31 | + |U 1 32 | + |U 1 34 | + 3|U 1 36 |, and by (13), (6), (12), and (14),…”

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