Summary: ``Let $G$ be a graph. A mapping $c\colon V(G)\to\{0,1,\dots,t-1\}$ is called a $t$-coloring of $G$, and $c$ is a proper coloring if $c(x)\neq c(y)$ whenever $x$ and $y$ are adjacent vertices in $G$. The (neighborhood) color code of vertex $v$ with respect to $c$, denoted $v^c$, is a $t$-tuple whose $k$th component $v^c_k$, $k=0,1,\dots,t-1$, is the number of times color $k$ is used among the neighbors of $v$. An irregular $t$-coloring of $G$ is a proper coloring $c$ such that for every pair of distinct vertices $x,y\in V_G$, $c(x)=c(y)$ implies $x^c\neq y^c$. The irregular chromatic number of graph $G$, denoted $x_{\rm ir}(G)$, is the smallest $t$ for which there exists an irregular $t$-coloring. This paper determines $\chi_{\rm ir}(C_n)$ and $\chi_{\rm ir}(P_n)$, which extends results of Okamoto, Radcliffe, and Zhang. Finally $\chi_{\rm ir}(K_m\times K_n)$ is also determined.''