On a conjecture concerning the property of chromatic polynomials with negative variables
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911527768948736 |
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| author | Yang, Yan |
| author_facet | Yang, Yan |
| contents | Let $G$ be a graph of order $n$ and $P(G,x)$ be the chromatic polynomial of $G$. Dong, Ge, Gong, Ning, Ouyang, and Tay (J. Graph Theory 96(2021) 343) conjectured that $\frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0$ holds for all $k \geq 2$ and $x \in (-\infty, 0)$. We prove this conjecture for all $k \geq 2 $ and $ x\leq -6.66Δk $, in which $Δ$ is the maximum degree of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07510 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a conjecture concerning the property of chromatic polynomials with negative variables Yang, Yan Combinatorics 05C31 Let $G$ be a graph of order $n$ and $P(G,x)$ be the chromatic polynomial of $G$. Dong, Ge, Gong, Ning, Ouyang, and Tay (J. Graph Theory 96(2021) 343) conjectured that $\frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0$ holds for all $k \geq 2$ and $x \in (-\infty, 0)$. We prove this conjecture for all $k \geq 2 $ and $ x\leq -6.66Δk $, in which $Δ$ is the maximum degree of $G$. |
| title | On a conjecture concerning the property of chromatic polynomials with negative variables |
| topic | Combinatorics 05C31 |
| url | https://arxiv.org/abs/2603.07510 |