  The temperature dependence of electron spin resonance (ESR) was studied in
$EuFe_{2-x}Co_{x}As_2 $ (x = 0.0, 0.067, 0.1, 0.2, 0.25, 0.285, 0.35, 0.4 and
0.5). The ESR spectrum of all the samples indicates that the g factor and
peak-to-peak linewidth strongly depend on the temperature. Moreover, the
peak-to-peak linewidth shows the Korringa behavior, indicating an exchange
coupling between the conduction electrons and the $Eu^{2+}$ions. The linewidth,
g factor and the integrate ESR intensity show anomalies at the temperature of
the spin-density-wave (SDW). The linewidth below the SDW transition does not
rely on the temperature. This gives the evidence of the gap opening at the
$T_{SDW}$. The slope of the linewidth is closely associated to $T_{SDW}$and
$T_C$. This exotic behavior may be related to the nesting of the Fermi surface.
