${\color{Red}B\color{Red}R \color{Red}O\color{Red}O\color{Red}K \,\,\text{THEOREM -:}}$-:$\text{Chromatic Number of anyCONNECTED graph having all its vertex atmost degree }\Delta \text{can be no greater than} \Delta \text{except complete graph and cycle graph}$
Petersen_graph is not a complete graph .It is connected and every vertex atmost $3$ degree so , according to Brook theorem ,
chromatic number of petersen graph=$3$