A coloring of a graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} is a partition {V1,V2,…,Vk}\documentclass[12pt]{minimal}
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\begin{document}$$\{V_1, V_2, \ldots , V_k\}$$\end{document} of V into independent sets or color classes. A vertex v∈Vi\documentclass[12pt]{minimal}
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\begin{document}$$v\in V_i$$\end{document} is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj\documentclass[12pt]{minimal}
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\begin{document}$$V_j$$\end{document} for every j<i\documentclass[12pt]{minimal}
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\begin{document}$$j<i$$\end{document}. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the Grundy number Γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma (G)$$\end{document} 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 {P4,C4}\documentclass[12pt]{minimal}
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\begin{document}$$\{P_{4}, C_4\}$$\end{document}-free graph by supporting a conjecture of Zaker, which says that Γ(G)≥δ(G)+1\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma (G)\ge \delta (G)+1$$\end{document} for any C4\documentclass[12pt]{minimal}
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\begin{document}$$C_4$$\end{document}-free graph G.
