Supersaturation for Subgraph Counts

The classic extremal problem is that of computing the maximum number of edges in an $F$-free graph. In the case where $F=K_{r+1}$, the extremal number was determined by Turan. Later results, known as supersaturation theorems, proved that in a graph containing more edges than the extremal number, there must also be many copies of $K_{r+1}$. Alon and Shikhelman introduced a broader class of problems asking for the maximum number of copies of a graph $T$ in an $F$-free graph. In this paper, we determine some of these generalized extremal numbers and prove supersaturation results for them.

[1]  David R. Wood,et al.  On the Maximum Number of Cliques in a Graph , 2006, Graphs Comb..

[2]  David Galvin,et al.  Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree , 2014, J. Graph Theory.

[3]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[4]  M. Simonovits,et al.  On the number of complete subgraphs of a graph II , 1983 .

[5]  Bernard Lidický,et al.  Maximizing five-cycles in Kr-free graphs , 2020, Eur. J. Comb..

[6]  P. Erdös,et al.  On the structure of linear graphs , 1946 .

[7]  Ruth Luo,et al.  The maximum number of cliques in graphs without long cycles , 2017, J. Comb. Theory B.

[8]  Ki Hang Kim,et al.  On a problem of Turán , 1983 .

[9]  B. Bollobás.  On complete subgraphs of different orders , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  Benny Sudakov,et al.  Maximizing the Number of Independent Sets of a Fixed Size , 2015, Comb. Probab. Comput..

[11]  Miklós Simonovits,et al.  On the maximal number of certain subgraphs inKr-free graphs , 1991, Graphs Comb..

[12]  Paul Erdös,et al.  On a theorem of Rademacher-Turán , 1962 .

[13]  Christian Reiher,et al.  The clique density theorem , 2012, 1212.2454.

[14]  J. Sheehan,et al.  On the number of complete subgraphs contained in certain graphs , 1981, J. Comb. Theory, Ser. B.

[15]  Raphael Yuster,et al.  A Tura'n Type Problem Concerning the Powers of the Degrees of a Graph , 2000, Electron. J. Comb..

[16]  Noga Alon,et al.  Many T copies in H-free graphs , 2014, Electron. Notes Discret. Math..

[17]  M. Aigner,et al.  Proofs from "The Book" , 2001 .

[18]  Anastasia Halfpap,et al.  On supersaturation and stability for generalized Turán problems , 2019, J. Graph Theory.

[19]  Vladimir Nikiforov,et al.  The number of cliques in graphs of given order and size , 2007, 0710.2305.

[20]  Ervin Györi,et al.  The maximum number of Pℓ copies in Pk-free graphs , 2019, Discret. Math. Theor. Comput. Sci..

[21]  Zoltán Füredi,et al.  New Asymptotics for Bipartite Turán Numbers , 1996, J. Comb. Theory, Ser. A.

[22]  Frank Harary,et al.  Graph Theory , 2016 .

[23]  Béla Bollobás,et al.  Degree Powers in Graphs with Forbidden Subgraphs , 2004, Electron. J. Comb..

[24]  V. Nikiforov.  Graphs with many r‐cliques have large complete r‐partite subgraphs , 2007, math/0703554.

[25]  R. A. R. A Z B O R O V.  On the minimal density of triangles in graphs , 2008 .

[26]  Alexander A. Razborov,et al.  On the Minimal Density of Triangles in Graphs , 2008, Combinatorics, Probability and Computing.

[27]  Béla Bollobás,et al.  Extremal problems in graph theory , 1977, J. Graph Theory.

[28]  Oleg Pikhurko,et al.  Supersaturation problem for color-critical graphs , 2017, J. Comb. Theory, Ser. B.