Disjoint isomorphic balanced clique subdivisions.

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 C k 2 has a subdivision of K k , 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 K k -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 K k. 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 K k -subdivision. This gives the right order of magnitude of the optimal d (k) needed in Thomassen's conjecture. Since a balanced K m k -subdivision trivially contains m vertex-disjoint isomorphic K k -subdivisions, this also confirms Verstraëte's conjecture in a strong sense. [ABSTRACT FROM AUTHOR]