Pavez-Signé’s question on similarly sized paths in spanning H-subdivisions

For a fixed graph $H$ with $h\ge 3$ edges and no isolated vertices, determine whether every sufficiently large $n$ -vertex graph $G$ satisfying the relevant minimum-degree hypothesis for a spanning $H$ -subdivision contains a spanning $H$ -subdivision whose replacement-path lengths $\ell_1,\ldots,\ell_h$ are nearly equal; asymptotically, this asks whether, for every $\varepsilon>0$ , one can require $\max_{1\le i,j\le h}|\ell_i-\ell_j|\le \varepsilon n$ while $\sum_{i=1}^h\ell_i=n-|V(H)|+h$ .

Progress summary

Partially solved

A new dense-graph theorem lets subdivision paths have prescribed, nearly equal lengths, but the general question remains open.

Pavez-Signé asked whether spanning subdivisions can require their replacement paths to have similar lengths. The question is not settled for arbitrary graphs, subdivision graphs, or length patterns.

Known results

Pavez-Signé: nearly balanced spanning $K_t$ -subdivisions for $t=O(\sqrt n)$ when $\delta(G)\ge(1+o(1))n/2$ .
Lee: the spanning-subdivision conjecture was solved using absorption, but the resulting subdivision may contain one path much longer than the others.
Related dense and pseudorandom results establish nearly balanced spanning clique subdivisions, but not arbitrary prescribed lengths.

August 2026 prescribed-length theorem

Wang, Zhilan, Wei, Shuo, Yan, and Jin prove that, for fixed $h\ge3$ , sufficiently large graphs with $\delta(G)\ge n/2+\lfloor h/3\rfloor$ contain spanning subdivisions of every $h$ -edge graph $H$ with no isolated vertices, for admissible lengths $\ell_i\ge4$ whose short-path contribution is at most $\beta n$ . The paths have exactly the prescribed lengths, hence can be nearly equal. A construction also shows the minimum-degree condition needs a linear additive term in general.

Current status (as of August 2026): A substantial dense-graph regime is settled, while Pavez-Signé’s general question remains open.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Spanning H-subdivisions with Prescribed Path Lengths — arXiv — Wang, Zhilan, Wei, Shuo, Yan, Jin

extremal-graph-theory graph-embeddings

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