Disjoint isomorphic balanced clique subdivisions




Fernández, Irene Gil
Hyde, Joseph
Liu, Hong
Pikhurko, Oleg
Wu, Zhuo

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Journal of Combinatorial Theory, Series B


A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least Ck2 has a subdivision of Kk, the complete graph on k vertices. We study two directions extending this result. • Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a Kk-subdivision. • Thomassen conjectured that for each k∈N there is some d=d(k) such that every graph with average degree at least d contains a balanced subdivision of Kk. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on d(k) remains open. In this paper, we show that a quadratic lower bound on average degree suffices to force a balanced Kk-subdivision. This gives the right order of magnitude of the optimal d(k) needed in Thomassen's conjecture. Since a balanced Kmk-subdivision trivially contains m vertex-disjoint isomorphic Kk-subdivisions, this also confirms Verstraëte's conjecture in a strong sense.



Balanced subdivisions, Average degree, Expander graphs, Cliques


Fernández, I. G., Hyde, J., Liu, H., Pikhurko, O., & Wu, Z. (2023). Disjoint isomorphic balanced clique subdivisions. Journal of Combinatorial Theory, Series B, 161, 417-436. https://doi.org/10.1016/j.jctb.2023.03.002