Summary: A coloring of a graph \(G=(V,E)\) is a partition \(\{V_1,V_2,\dots,V_k\}\) 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 \(G\).