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Multiplicities of subgraphs

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Abstract

A former conjecture of Burr and Rosta [1], extending a conjecture of Erdős [2], asserted that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graphG which are monochromatic is at least the proportion found in a random colouring. It is now known that the conjecture fails for some graphsG, includingG=K p forp≥4.

We investigate for which graphsG the conjecture holds. Our main result is that the conjecture fails ifG containsK 4 as a subgraph, and in particular it fails for almost all graphs.

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References

  1. S. A. Burr andV. Rosta: On the Ramsey multiplicities of graph problems and recent results.J. Graph Theory,4, (1980) 347–361.

    Google Scholar 

  2. P. Erdős: On the number of complete subgraphs contained in certain graphs,Publ. Math. Inst. Hung. Acad. Sci., VII, Ser. A3 (1962) 459–464.

    Google Scholar 

  3. P. Erdős andJ. W. Moon: On subgraphs of the complete bipartite graph,Canad. Math. Bull. 7 (1964) 35–39.

    Google Scholar 

  4. P. Erdős andM. Simonovits: Supersaturated graphs and hypergraphs,Combinatorica 3 (2) (1983) 181–192.

    Google Scholar 

  5. F. Franek andV. Rödl: 2-Colorings of complete graphs with a small number of monochromaticK 4 subgraphs,Discrete Mathematics,114, (1993) 199–203.

    Google Scholar 

  6. G. Giraud: Sur le probléme de Goodman pour les quadrangles et la majoration des nombre de Ramsey,J. Combin. Theory, Series B,27, (1979) 237–253.

    Google Scholar 

  7. A. W. Goodman: On sets of acquaintances and strangers at any party,Amer. Math. Monthly,66 (1959), 778–783.

    Google Scholar 

  8. C. N. Jagger: Ph.D. thesis, (in preparation).

  9. G. Lorden: Blue-empty chromatic graphs,Amer. Math. Monthly,69 (1962), 114–120.

    Google Scholar 

  10. A. F. Sidorenko: Cycles in graphs and functional inequalities,Mathematical Notes,46 (1989), 877–882.

    Google Scholar 

  11. A. F. Sidorenko: Extremal problems for finite sets,Bolyai Society Mathematical Studies,3 (1994), 425–457.

    Google Scholar 

  12. A. F. Sidorenko: A correlation inequality for bipartite graphs,Graphs and Combinatorics,9 (1993), 201–204.

    Google Scholar 

  13. A. F. Sidorenko: Inequalities for functionals generated by bipartite graphs,Discrete Math. Appl.,2 (5) (1992), 489–504.

    Google Scholar 

  14. A. F. Sidorenko: Extremal problems in graph theory and inequalities in functional analysis,Proc. Soviet Seminar on Discrete Math. Appl. Moscow Univ Press, 1986, 99–105 (in Russian).

  15. A. G. Thomason: A disproof of a conjecture of Erdős in Ramsey Theory,J. London Math. Soc.,39 (2) (1989), 246–255.

    Google Scholar 

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, 16, Mill Lane, CB2 1SB, Cambridge, England

    Chris Jagger & Andrew Thomason

  2. Department of Mathematics, Faculty of Nuclear Science, Trojanova 13, 120 00, Prague, Czech Republic

    Pavel Šťovíček

Authors
  1. Chris Jagger
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  2. Pavel Šťovíček
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  3. Andrew Thomason
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Jagger, C., Šťovíček, P. & Thomason, A. Multiplicities of subgraphs. Combinatorica 16, 123–141 (1996). https://doi.org/10.1007/BF01300130

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  • DOI: https://doi.org/10.1007/BF01300130

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