 
Summary: Com,,,,,poIMy ~lf>lh"",",tic'
Volume 531, 2010
The proportion of various graphs in graphdesigns
Richard 1\1. \\Tilson
AIJ~TIIV"l'. Let 9 be a family of simple graph,. i\ 9deHign on 1l points is
a decomposition of the "dges of r(n into copi"" of graphs in g, Tit case ,bat
9 consists of compkte graphs r(" wi,h k in some sci j( of pOHitiv" integel'H,
SL!ch a 9Jesign is calkJ a pairwise balanceJ design (Pl3D) 011 n pointH witb
block sizes from rc Here We arC conceT/",d with the posoible pl'OporLioJlS of
the nllmbers of copies of graphH G f:: 9 that appear in decompositions for large
n. \Ve ext(md a n,snh ot Colbomn "nd Riid] on I'GDH to 9d"signs, '1nd giv" a
fL!rtlwr result 011 (;he pOHHible llllmbers of COpi"H o[ G in a (;;J""ign cont'li11illg
eacb vertex of the compldc' graph r(n,
1.. Introduction
Por a positivp intpgpr n and a sel K of posithT integers, a 2(71" K, 1) desWfl
cOllHist.s of a sd X of n poinls and a family A of HuhHds 0(' X, called blocks, so tbat
IAI E K for pvery A E .4, and pvery f;ubset {x, /1} of Lwo pointf; in X is conlained in
a unique member of' A. These may also be called pwin1!ise balanced designs (PBDs)
with block sizes iII K.
We use a(K) for t,bp gcd (great,esl common dhcisor) of {k  1 k F 1q and
