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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.04990 |
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| _version_ | 1866914077832380416 |
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| author | Cordero-Michel, Narda Olsen, Mika |
| author_facet | Cordero-Michel, Narda Olsen, Mika |
| contents | Given a digraph $D$ with no loops, the \textit{dicoloring graph} of $D$, denoted by $\mathcal{D}_k(D)$, is the graph whose vertices are the acyclic $k$-colorings of $D$ and two colorings are adjacent in $\mathcal{D}_k(D)$ if they differ in color on exactly one vertex. In this paper, we prove that there is no expression $ϕ(\vecχ)$ in terms of the dichromatic number $\vecχ$, such that the graph $\mathcal{D}_k(D)$ is connected for all graphs $D$ and integers $k\geq ϕ(\vecχ)$. We give conditions for the dicoloring graph of two infinite families of circulant tournaments to be connected, and we provide upper bounds for its diameter. In particular, for the Payley tournament $\vec{C}_{7}(1,2,4)$, also known as $ST_7$, we prove that $\mathcal{D}_k(\vec{C}_{7}(1,2,4))$ is connected and has diameter 8, for each $k\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Redicoloring some classes of circulant tournaments Cordero-Michel, Narda Olsen, Mika Combinatorics 05C15, 05C20 Given a digraph $D$ with no loops, the \textit{dicoloring graph} of $D$, denoted by $\mathcal{D}_k(D)$, is the graph whose vertices are the acyclic $k$-colorings of $D$ and two colorings are adjacent in $\mathcal{D}_k(D)$ if they differ in color on exactly one vertex. In this paper, we prove that there is no expression $ϕ(\vecχ)$ in terms of the dichromatic number $\vecχ$, such that the graph $\mathcal{D}_k(D)$ is connected for all graphs $D$ and integers $k\geq ϕ(\vecχ)$. We give conditions for the dicoloring graph of two infinite families of circulant tournaments to be connected, and we provide upper bounds for its diameter. In particular, for the Payley tournament $\vec{C}_{7}(1,2,4)$, also known as $ST_7$, we prove that $\mathcal{D}_k(\vec{C}_{7}(1,2,4))$ is connected and has diameter 8, for each $k\geq 3$. |
| title | Redicoloring some classes of circulant tournaments |
| topic | Combinatorics 05C15, 05C20 |
| url | https://arxiv.org/abs/2510.04990 |