Sampling Colorings with Fixed Color Class Sizes
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914379880988672 |
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| author | Kuchukova, Aiya Perkins, Will Povill, Xavier |
| author_facet | Kuchukova, Aiya Perkins, Will Povill, Xavier |
| contents | In 1970 Hajnal and Szemerédi proved a conjecture of Erdös that for a graph with maximum degree $Δ$, there exists an equitable $Δ+1$ coloring; that is a coloring where color class sizes differ by at most $1$. In 2007 Kierstand and Kostochka reproved their result and provided a polynomial-time algorithm which produces such a coloring. In this paper we study the problem of approximately sampling uniformly random equitable colorings. A series of works gives polynomial-time sampling algorithms for colorings without the color class constraint, the latest improvement being by Carlson and Vigoda for $q\geq 1.809 Δ$. In this paper we give a polynomial-time sampling algorithm for equitable colorings when $q> 2Δ$. Moreover, our results extend to colorings with small deviations from equitable (and as a corollary, establishing their existence). The proof uses the framework of the geometry of polynomials for multivariate polynomials, and as a consequence establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08259 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sampling Colorings with Fixed Color Class Sizes Kuchukova, Aiya Perkins, Will Povill, Xavier Combinatorics Data Structures and Algorithms In 1970 Hajnal and Szemerédi proved a conjecture of Erdös that for a graph with maximum degree $Δ$, there exists an equitable $Δ+1$ coloring; that is a coloring where color class sizes differ by at most $1$. In 2007 Kierstand and Kostochka reproved their result and provided a polynomial-time algorithm which produces such a coloring. In this paper we study the problem of approximately sampling uniformly random equitable colorings. A series of works gives polynomial-time sampling algorithms for colorings without the color class constraint, the latest improvement being by Carlson and Vigoda for $q\geq 1.809 Δ$. In this paper we give a polynomial-time sampling algorithm for equitable colorings when $q> 2Δ$. Moreover, our results extend to colorings with small deviations from equitable (and as a corollary, establishing their existence). The proof uses the framework of the geometry of polynomials for multivariate polynomials, and as a consequence establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings. |
| title | Sampling Colorings with Fixed Color Class Sizes |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2603.08259 |