In this paper we report the implementation of low resistance ohmic contacts to p-type ZnSSe and ZnMgSSe which involves the injection of holes from heavily doped ZnTe into an adjacent alloy layer(s) via graded band gap regions. Temperature-dependent Hall effect measurements on nitrogen-doped Zn(S,Se) and (Zn,Mg)(S,Se) were performed and the activation energy of nitrogen acceptors was obtained. With the use of this graded contact, room-temperature continuous-wave laser diode operation has been achieved at a threshold voltage of 5.8 V in a ZnCdSe/ZnSSe/ZnMgSSe separate confinement heterostructure.