Extremal density for subdivisions with length or sparsity constraints
Year: 2025
Type: article
Abstract: Given a graph H , a balanced subdivision of H is obtained by replacing all edges of H with internally disjoint paths of the same length. In this paper, we prove that for any graph H , a linear-in- e ( H ) bound on average degree guarantees a balanced H -subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever H is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs H below this density, its subdivisions are forcible by a sublinear-in- e ( H ) bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph H with sublogarithmic density are forcible by a sublinear-in- e ( H ) bound on average degree, provided that H satisfies some additional separability condition.
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,150
Funders: National Research Foundation, National Natural Science Foundation of China, China Postdoctoral Science Foundation, National Research Foundation of Korea, China Scholarship Council, Natural Science Foundation of Shandong Province, Ministry of Science and ICT, South Korea, National Key Research and Development Program of China
OpenAlex ID: W4416431302