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An extremal function for contractions of graphs

  • Andrew Thomason (a1)
Abstract

The function c(p) is defined for positive integers p ≥ 4 by

where > denotes contraction. Random graph examples show

In 1968 Mader showed that c(p) ≤ 8(p − 2) [log2 (p − 2)] and more recently Kostochka showed that p√(log p) is the correct order for c(p). We present a simple argument showing c(p) ≤ 2.68p √(log2p)(l + ο(l)).

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References
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[1]Bollobás, B.. Extremal Graph Theory (Academic Press, 1978).
[2]Bollobás, B., Catlin, P. and Erdös, P.. Hadwiger's conjecture is true for almost every graph. European J. Combin. 1 (1980), 195199.
[3]Kostochka, A. V.. A lower bound for the Hadwiger number of a graph as a function of the average degree of its vertices. Discret. Analyz. Novosibirsk 38 (1982), 3758.
[4]Mader, W.. Homomorphiesätze für Graphen. Math. Ann. 178 (1968), 154168.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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