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An extremal function for contractions of graphs. (English) Zbl 0551.05047
For finite graphs $$G$$ and $$H$$ the author writes $$G>H$$ when $$H$$ may be obtained from $$G$$ by a sequence of deletions and contractions of edges. For integers $$p\geq 4$$, $$c(p)$$ is defined to be $$\inf \{e(G)/v(G): G>K_ p\}$$. “Since a random graph of order $$n$$ with edges chosen with probability $$1-q$$ almost certainly does not contract to a $$K_ p$$ where $$p=n\{\log (1/q)/\log (n)\}^{\frac{1}{2}}\{1+o(1)\}$$ [B. Bollobás, P. Catlin, and P. Erdős, Eur. J. Comb. 1, 195–199 (1980; Zbl 0457.05041)] …we see, on taking $$q=0.284$$, that $$c(p)\geq 0.265p \log_ 2p$$ for large $$p$$.” W. Mader [Math. Ann. 178, 154–168 (1968; Zbl 0165.57401)] has shown that $$c(p)\leq 8(p-2)\log_ 2(p-2)$$. The author improves an upper bound of A. V. Kostochka [Metody Diskretn. Anal. 38, 37–58 (1982; Zbl 0544.05037)] (that $$c(p)<324\sqrt{\log_ 2p})$$ by proving that $$c(p)\leq 2.68p\sqrt{\log_ 2p}$$ for large $$p$$.
Reviewer: W. G. Brown

##### MSC:
 05C35 Extremal problems in graph theory 60C05 Combinatorial probability 05C80 Random graphs (graph-theoretic aspects)
##### Keywords:
contractions of edges; Hadwiger number; average degree
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##### References:
 [1] DOI: 10.1007/BF01350657 · Zbl 0165.57401 · doi:10.1007/BF01350657 [2] Bollob?s, Extremal Graph Theory (1978) [3] Bollob?s, European J. Combin. 1 pp 195– (1980) · Zbl 0457.05041 · doi:10.1016/S0195-6698(80)80001-1 [4] Kostochka, Discret. Analyz. Novosibirsk 38 pp 37– (1982)
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