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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.13221 |
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| _version_ | 1866913044810956800 |
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| author | Ning, Bo Yang, Yan |
| author_facet | Ning, Bo Yang, Yan |
| contents | Let \( G \) be a graph of order \( n \) with maximum degree $Δ$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)/x^n$. Dong proved that $(x-1)^nP(G,x)\geq x^nP(G,x-1)$ for all real $x\geq n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)/P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\geq 36Δ^{3/2}$. We improve this bound by proving that it also holds for all real $x\geq 10Δ^{3/2}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that \( \frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0 \) for all \( k \geq 2 \) and \( x \in (-\infty, 0) \). We establish this conjecture for all \( k \geq 2 \) and \( x\leq -3.01Δk \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On derivatives and higher-order derivatives of chromatic polynomials Ning, Bo Yang, Yan Combinatorics Let \( G \) be a graph of order \( n \) with maximum degree $Δ$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)/x^n$. Dong proved that $(x-1)^nP(G,x)\geq x^nP(G,x-1)$ for all real $x\geq n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)/P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\geq 36Δ^{3/2}$. We improve this bound by proving that it also holds for all real $x\geq 10Δ^{3/2}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that \( \frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0 \) for all \( k \geq 2 \) and \( x \in (-\infty, 0) \). We establish this conjecture for all \( k \geq 2 \) and \( x\leq -3.01Δk \). |
| title | On derivatives and higher-order derivatives of chromatic polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.13221 |