Disjoint isomorphic balanced clique subdivisions

Date

2023

Authors

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

Journal Title

Journal ISSN

Volume Title

Publisher

Journal of Combinatorial Theory, Series B

Abstract

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.

Description

Keywords

Balanced subdivisions, Average degree, Expander graphs, Cliques

Citation

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