More bounds for the Grundy number of graphs

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 $\mathrm{\Gamma }(G)$ 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 ${P_{4}, C_4}$-freegraphbysupportingaconjectureofZaker,whichsaysthat$\Gamma(G)\geq \delta(G)+1$forany$C_4$-freegraph$G$.