Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Saturation Numbers of Books. | Saturation Numbers of Books Guantao Chen Dept. of Math. and Stat. Georgia State University Atlanta GA 30303 gchen@ Ralph J. Faudree Dept. of Math. Sciences University of Memphis Memphis TN 38152 rfaudree@ Ronald J. Gould Dept. of Math. and Computer Science Emory University Atlanta GA 30322 rg@ Submitted Oct 17 2007 Accepted Sep 5 2008 Published Sep 15 2008 Mathematics Subject Classifications 05C35 Abstract 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 u 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 sat Bp n 2 p 1 n - 1 - l p mi p I e n p Á I Á I I Á 1 z where e n p p f p n p 2 0 mod 2 provided n p3 p. 0 otherwise Moreover we show that sat Bb p n 1 p 2b - 3 n - b 1 - lpm i p I ỡ n p b b - 1 b - 2 2 X I 2 I I 2 I z where e n p b p if p n p 2 b 0 mod 2 provided n 4 p 2b b. 0 otherwise The work was partially supported by NSF grant DMS-0070514 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R118 1 1 Introduction In this paper we consider only graphs without loops or multiple edges. For terms not defined here see 1 . We use A B to define A as B. Let G be a graph with vertex set V G and edge set E G . We call n G V G I the order of G and G E G I the size of G. For any v 2 V G let N v w vw 2 Eg be the neighborhood of v N v N v u vg be the closed neighborhood of v and d v N v be the degree of v. Furthermore if U c V G we will use Ui to denote the subgraph of G induced by U. Let Nu v N v U and du v u v . The complement of G is denoted by G. Let G and H be graphs. We say that G is H-saturated if H is not a subgraph of G but for any edge uv in G H is a subgraph of G uv. For a fixed integer