This paper describes an improvement in the upper bound for the magnitude of a coefficient of a term in the chromatic polynomial of a general graph. If $a_r$ is the coefficient of the $q^r$ term in the chromatic polynomial $P(G,q)$, where $q$ is the number of colors, then we find $a_r \le { e \choose v-r} - { e-g+2 \choose v-r-g+2} + { e-k_g-g+2 \choose v-r-g+2} - \sum _{ n=1}^{ k_g-\ell_g}\sum_{ m=1}^{ \ell_g-1} { e-g+1-n-m \choose v-r-g} - \delta_{ g,3}\sum_{ n=1}^{ k_g+\ell_{ g+1}^*-\ell_g} { e-\ell_g-g+1-n \choose v-r-g}$, where $k_g$ is the number of circuits of length $g$ and $\ell_g$ and $\ell_{ g+1}^*$ are certain numbers defined in the text.
Categories: math.CO