## 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.

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## Acknowledgements

The authors would like to thank the anonymous referee for their careful reading of the paper and their helpful comments which improved the presentation of this paper. We want to mention that Fernández, Hyde, Liu, Pikhurko and Wu [6] obtained a similar result regarding Theorem 1.2. The results in two papers are finished independently.

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Bingyu Luan and Guanghui Wang were supported by Natural Science Foundation of China (11871311), Young Taishan Scholars Program and seed fund program for international research cooperation of Shandong University, Donglei Yang: Supported by the China Postdoctoral Science Foundation (2021T140413), Natural Science Foundation of China (12101365) and Natural Science Foundation of Shandong Province (ZR2021QA029).

## A Dependent random choice

### A Dependent random choice

The following lemma can be regarded as a bipartite version of dependent random choice, of which the proof follows from that of Fox and Sudakov [7].

### Lemma A.1

Given integers \(a, t, n_1,n_2, c, r\) and a constant \(\alpha >0\), let \(G=(V_1,V_2,E)\) be a bipartite graph such that \(|V_1|=n_1\), \(|V_2|=n_2\) and \(|E|\ge \alpha n_1n_2\). If it holds that

then there exists a set \(A_0\subseteq V_1\) of size at least *a* such that every *r*-subset of \(A_0\) has at least *c* common neighbours in \(V_2\).

### Proof

Pick a set of *t* vertices of \(V_2\) uniformly at random with repetition, say \(b_1,b_2,\cdots ,b_t\). Let *A* denote the set of common neighbours for all vertices \(b_i\) and \(X=|A|\). By linearity of expectation,

By the convexity of the function \(f(x)=x^t\), we have

Let *Y* denote the random variable counting the number of *r*-subsets in *A* with fewer than *c* common neighbours in \(V_2\). Therefore, the probability that a randomly chosen \(b_i\) is one of the common neighbours of such an *r*-set is at most \(\frac{c}{n_2}\). Hence, since we made random choices of \(b_i\) uniformly and independently, the probability that such an *r*-tuple be contained in *A* is at most \(\left( \frac{c}{n_2}\right) ^t\). As there are at most \(\left( {\begin{array}{c}|V_1|\\ r\end{array}}\right) \) subsets of size *r*, it follows that

Again, by linearity of expectation, it holds that

Hence there exists a choice of *A* for which \(X-Y\ge a\). Delete one vertex from each subset *r*-subset of *A* with fewer than *c* common neighbours and let \(A_0\) be the remaining subset of *A*. Thus, \(|A_0|\ge a\) and every *r*-subset of \(A_0\) has at least *c* common neighbours. \(\square \)

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Luan, B., Tang, Y., Wang, G. *et al.* Balanced Subdivisions of Cliques in Graphs.
*Combinatorica* **43**, 885–907 (2023). https://doi.org/10.1007/s00493-023-00039-2

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DOI: https://doi.org/10.1007/s00493-023-00039-2