Triangles in H-free graphs
نویسنده
چکیده
For two graphs T and H and for an integer n, let ex(n, T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T = K2 is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the case T = K3, which reveals several interesting phenomena. Three representative results are: (i) ex(n,K3, H) ≤ c(H)n iff the 2-core of H is a friendship graph, (ii) For any fixed s ≥ 2 and t ≥ (s− 1)! + 1, ex(n,K3,Ks,t) = Θ(n3−3/s), and (iii) ex(n,K3, C5) ≤ (1 + o(1)) √ 3 2 n . The last statement improves (slightly) an estimate of Bollobás and Győri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques.
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