Balanced Subdivisions of Cliques in Graphs

Given a graph H, a balanced subdivision of H is a graph obtained from H by sub-dividing every edge the same number of times. In 1984, Thomassen conjectured that for each integer k = 1, high average degree is sufficient to guarantee a balanced sub-division of K-k. Recently, Liu and Montgomery resolved this conjecture.We give an optimal estimate up to an absolute constant factor by showing that there exists c > 0 such that for sufficiently large d, every graph with average degree at least d contains a balanced subdivision of a clique with at least cd(1/2) vertices. It also confirms a conjec-ture from Verstraete: every graph of average degree cd(2), for some absolute constant c > 0, contains a pair of disjoint isomorphic subdivisions of the complete graph K-d. We also prove that there exists some absolute c > 0 such that for sufficiently large d, every C-4-free graph with average degree at least d contains a balanced subdivision of the complete graph K-cd, which extends a result of Balogh, Liu and Sharifzadeh.