Balanced Subdivisions of Cliques in Graphs

Given a graph H , a balanced subdivision of H is a graph obtained from H by subdividing 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 subdivision 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 conjecture from Verstraëte: 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.