New Bounds for Chromatic Polynomials and Chromatic Roots

If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $π(G,x)$, is at most $x(x-1)\cdots (x-(k-1))x^{n-k} = (x)_{\downarrow k}x^{n-k}$ for every $x\in \mathbb{N}$. We improve here this bound by showing that $π(G,x) \leq (x)_{\downarrow k} (x-1)^{Δ(G)-k+1} x^{n-1-Δ(G)}$ for every $x\in \mathbb{N},$ where $Δ(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\geq 4$ then $π(G,x)$ is at most $(x)_{\downarrow k}(x-1)^{n-k}$ for every real $x\geq n-2+\left( {n \choose 2} -{k \choose 2}-n+k \right)^2$ (it had been previously conjectured that this inequality holds for all $x \geq k$). Finally, we provide an upper bound on the moduli of the chromatic roots that is an improvment over known bounds for dense graphs.

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