More bounds for the Grundy number of graphs

Abstract: A coloring of a graph $G=(V,E)$ is a partition ${V_1, V_2, \ldots, V_k}$ of $V$ into independent sets or color classes. A vertex $v\in V_i$ is a Grundy vertex if it is adjacent to at least one vertex in each color class $V_j$ for every $j<i$. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the Grundy number $\Gamma(G)$ of a graph $G$ is the maximum number of colors in a Grundy coloring. We provide two new upper bounds on Grundy number of a graph and a stronger version of the well-known Nordhaus-Gaddum theorem. In addition, we give a new characterization for a ${P_{4}, C_4}$-free graph by supporting a conjecture of Zaker, which says that $\Gamma(G)\geq \delta(G)+1$ for any $C_4$-free graph $G$.