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On the maximal number of certain subgraphs inK r -free graphs

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Abstract

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) ≤ H(T 2(n)), whereT 2(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n ≥ max(m, r − 1), then there exists a complete (r − 1)-partite graphG* withn vertices such thatK(G) ≤ K(G*) holds for everyK r -free graphG withn vertices. In particular, in the class of allK r -free graphs withn vertices the complete balanced (r − 1)-partite graphT r−1(n) has the largest number of subgraphs isomorphic toK t (t < r),C 4,K 2,3. These generalize some theorems of Turán, Erdös and Sauer.

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Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, 1364, Budapest, Hungary

    Ervin Györi, János Pach & Miklós Simonovits

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  1. Ervin Györi
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  2. János Pach
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  3. Miklós Simonovits
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Dedicated to Paul Turán on his 80th Birthday

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Györi, E., Pach, J. & Simonovits, M. On the maximal number of certain subgraphs inK r -free graphs. Graphs and Combinatorics 7, 31–37 (1991). https://doi.org/10.1007/BF01789461

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  • DOI: https://doi.org/10.1007/BF01789461

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