TY - JOUR

T1 - On the number of certain subgraphs contained in graphs with a given number of edges

AU - Alon, Noga

PY - 1986/2

Y1 - 1986/2

N2 - 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(l, 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 s 1≧s 2≧...≧s r>r, provided (s 1-s r)2 1+s r-2 r. We further show that if H is a graph with k edges, then the ratio N(l, H)/l k tends to a finite limit as l→∞. This limit is non-zero iff H is a disjoint union of stars.

AB - 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(l, 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 s 1≧s 2≧...≧s r>r, provided (s 1-s r)2 1+s r-2 r. We further show that if H is a graph with k edges, then the ratio N(l, H)/l k tends to a finite limit as l→∞. This limit is non-zero iff H is a disjoint union of stars.

UR - http://www.scopus.com/inward/record.url?scp=51249173411&partnerID=8YFLogxK

U2 - 10.1007/BF02772673

DO - 10.1007/BF02772673

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AN - SCOPUS:51249173411

SN - 0021-2172

VL - 53

SP - 97

EP - 120

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -