Minimal H-factors and covers
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929719148019712 |
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| author | Federico, Lorenzo Danielsson, Joel Larsson |
| author_facet | Federico, Lorenzo Danielsson, Joel Larsson |
| contents | Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs $H'\subset G$, each isomorphic to H, that cover the vertices of G.
If G is the complete graph $K_n$ equipped with independent U(0,1) edge weights, what is the lowest total weight of an H-factor? This problem has previously been considered for e.g.\ $H=K_2$.
We show that if H contains a cycle, then the minimum weight is sharply concentrated around some $L_n = Θ(n^{1-1/d^*})$ (where $d^*$ is the maximum 1-density of any subgraph of H). Some of our results also hold for H-covers, where the copies of H are not required to be vertex-disjoint. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12184 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minimal H-factors and covers Federico, Lorenzo Danielsson, Joel Larsson Combinatorics Probability 05C80 Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs $H'\subset G$, each isomorphic to H, that cover the vertices of G. If G is the complete graph $K_n$ equipped with independent U(0,1) edge weights, what is the lowest total weight of an H-factor? This problem has previously been considered for e.g.\ $H=K_2$. We show that if H contains a cycle, then the minimum weight is sharply concentrated around some $L_n = Θ(n^{1-1/d^*})$ (where $d^*$ is the maximum 1-density of any subgraph of H). Some of our results also hold for H-covers, where the copies of H are not required to be vertex-disjoint. |
| title | Minimal H-factors and covers |
| topic | Combinatorics Probability 05C80 |
| url | https://arxiv.org/abs/2302.12184 |