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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.07783 |
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| _version_ | 1866916998129123328 |
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| author | Zhang, Menglong Ge, Gennian |
| author_facet | Zhang, Menglong Ge, Gennian |
| contents | A $K_4$-decomposition of a graph is a partition of its edges into $K_4$s. A fractional $K_4$-decomposition is an assignment of a nonnegative weight to each $K_4$ in a graph such that the sum of the weights of the $K_4$s containing any given edge is one. Formulating a nonlinear programming and reducing the number of variables slowly, we prove that every graph on $n$ vertices with minimum degree at least $\frac{31}{33}n$ has a fractional $K_4$-decomposition. This improves a result of Montgomery that the same conclusion holds for graphs with minimum degree at least $\frac{399}{400}n$. Together with a result of Barber, Kühn, Lo, and Osthus, this result implies that for all $\varepsilon> 0$, every large enough $K_4$-divisible graph on $n$ vertices with minimum degree at least $(\frac{31}{33}+\varepsilon)n$ admits a $K_4$-decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Progress towards generalized Nash-Williams' conjecture on $K_4$-decompositions Zhang, Menglong Ge, Gennian Combinatorics A $K_4$-decomposition of a graph is a partition of its edges into $K_4$s. A fractional $K_4$-decomposition is an assignment of a nonnegative weight to each $K_4$ in a graph such that the sum of the weights of the $K_4$s containing any given edge is one. Formulating a nonlinear programming and reducing the number of variables slowly, we prove that every graph on $n$ vertices with minimum degree at least $\frac{31}{33}n$ has a fractional $K_4$-decomposition. This improves a result of Montgomery that the same conclusion holds for graphs with minimum degree at least $\frac{399}{400}n$. Together with a result of Barber, Kühn, Lo, and Osthus, this result implies that for all $\varepsilon> 0$, every large enough $K_4$-divisible graph on $n$ vertices with minimum degree at least $(\frac{31}{33}+\varepsilon)n$ admits a $K_4$-decomposition. |
| title | Progress towards generalized Nash-Williams' conjecture on $K_4$-decompositions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.07783 |