Balanced Subdivisions of Cliques in Graphs

Abstract

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 $$k\ge 1$$ , 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.