A local coloring of a graph \(G\) is a function \(c: V(G) \to \mathbb{N}\) having the property that for each set \(S \subseteq V(G)\) with \(2 \leq |S| \leq 3\), there exist vertices \(u,v \in S\) such that \(|c(u) – c(v)| \geq m_S\), where \(m_S\) is the size of the induced subgraph \(\langle S\rangle\). The maximum color assigned by a local coloring \(c\) to a vertex of \(G\) is called the value of \(c\) and is denoted by \(\chi_\ell(c)\). The local chromatic number of \(G\) is \(\chi_\ell(G) = \min\{\chi_\ell(c)\}\), where the minimum is taken over all local colorings \(c\) of \(G\). If \(\chi_\ell(c) = \chi_\ell(G)\), then \(c\) is called a minimum local coloring of \(G\). The local coloring of graphs introduced by Chartrand et al. in \(2003\). In this paper, following the study of this concept, first an upper bound for \(\chi_\ell(G)\) where \(G\) is not complete graphs \(K_4\) and \(K_5\), is provided in terms of maximum degree \(\Delta(G)\). Then the exact value of \(\chi_\ell(G)\) for some special graphs \(G\) such as the cartesian product of cycles, paths and complete graphs is determined.