On the number of subgraphs of prescribed type of graphs with a given number of edges

AbstractAll graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For a graphH=〈V(H),E(H)〉 and forS ⊂V(H) defineN(S)={x ∈V(H):xy ∈E(H) for somey ∈S}. Define alsoδ(H)= max {|S| − |N(S)|:S ⊂V(H)},γ(H)=1/2(|V(H)|+δ(H)). For two graphsG, H letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also forl>0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We investigate the asymptotic behaviour ofN(l, H) for fixedH asl tends to infinity. The main results are:Theorem A.For every graph H there are positive constants c1, c2 such that {fx116-1}. Theorem B.If δ(H)=0then {fx116-2},where |AutH|is the number of automorphisms of H.(It turns out thatδ(H)=0 iffH has a spanning subgraph which is a disjoint union of cycles and isolated edges.)