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Summary: ISRAEL IOURNAL OF MATHEMATICS, Vol. 53, No. 1, 1986
ON THE NUMBER OF CERTAIN SUBGRAPHS
CONTAINED IN GRAPHS WITH
A GIVEN NUMBER OF EDGES
BY
NOGA ALON
SchoolofMathematicalSciences,TelAviv University,RamatAviv, Israel
ABSTRACT
All graphs considered are finite, undirected, with no loops, no multiple edges
and no isolated vertices. For two graphs G, H, let N(G, H) denote the number
of subgraphs of G isomorphic to H. Define also, for l >=0, N(I,H)=
max N(G,H), where the maximum is taken over all graphs G with l edges. We
determine N(l,H) precisely for all l ->0 when H is a disjoint union of two stars,
and also when H is a disjoint union of r > 3 stars, each of size s or s + 1, where
s ->_r. We also determine N(l,H) for sufficiently large l when H is a disjoint
union of r stars, of sizes st ->s2----"' ·~ s, > r, provided (S 1 -- S,)2< S131-Sr --2r.
We further show that if H is a graph with k edges, then the ratio N(l, H)/lk
tends to a finite limit as l ~ c,. This limit is non-zero iff H is a disjoint union of
stars.
1. Introduction
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