Extremal connectivity for topological cliques in bipartite graphs

Authors:
Daniela Kühn;Deryk Osthus
Affiliations:
School of Mathematics, Birmingham University, Edgbaston, Birmingham, UK;School of Mathematics, Birmingham University, Edgbaston, Birmingham, UK
Venue:
Journal of Combinatorial Theory Series B
Year:
2006

Citing 5
Cited 0

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Abstract

Let d(s) be the smallest number such that every graph of average degree d(s) contains a subdivision of Ks. So far, the best known asymptotic bounds for d(s) are (1 + o(1))9s2/64 ≤ d(s) ≤ (1 + o(1))s2/2. As observed by Łuczak, the lower bound is obtained by considering bipartite random graphs. Since with high probability the connectivity of these random graphs is about the same as their average degree, a connectivity of (1 + o(1))9s2/64 is necessary to guarantee a subdivided Ks. Our main result shows that for bipartite graphs this gives the correct asymptotics. We also prove that in the non-bipartite case a connectivity of (1 + o(1))s2/4 suffices to force a subdivision of Ks. Moreover, we slightly improve the constant in the upper bound for d(s) from 1/2 (which is due to Komlós and Szemerédi) to 10/23.