Upper Bounds on the Chromatic Polynomial of a Connected Graph with Fixed Clique Number

Abstract

Let $\mathcal {F}_{t}(n)$ denote the family of all connected graphs of order n with clique number t. In this paper, we present a new upper bound for the chromatic polynomial of a graph G in $\mathcal {F}_{t}(n)$ in terms of the clique number of G. Moreover, we also show that the conjecture proposed by Tomescu, which says that if $x\ge k\ge 4$ and G is a connected graph on n vertices with chromatic number k, then

$$\begin{aligned} P(G,x)\le (x)_{k} (x-1)^{n-k} \end{aligned}$$

holds under certain conditions, where $(x)_{k}=x(x-1)\cdots (x-k+1)$, such as the clique number $\omega (G)=k\ge 4$, $\omega (G)=k-1 \ (k\ge 7)$ with two $(k-1)$-cliques having at most $k-3$ vertices in common, or maximum degree $\triangle (G)=n-2$.