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Saturation numbers of books

By Guantao Chen, Ronald J. Gould and Ralph J. Faudree


A book Bp is a union of p triangles sharing one edge. This idea was extended to a generalized book Bb,p, which is the union of p copies of a Kb+1 sharing a common Kb. A graph G is called an H-saturated graph if G does not contain H as a subgraph, but G ∪ {xy} contains a copy of H, for any two nonadjacent vertices x and y. The saturation number of H, denoted by sat(H, n), is the minimum number of edges in G for all H-saturated graphs G of order n. We show that (p + 1)(n − 1) − sat(Bp, n) = 1 2 1 if p ≡ n − p/2 ≡ 0 mod 2 where θ(n, p) = 0 otherwise Moreover, we show that sat(Bb,p, n) = 1 2 where θ(n, p, b) = (p + 2b − 3)(n − b + 1)

Topics: 2
Year: 2010
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