I was wondering why some of the sahabas didn’t give oath to the Caliphate of 'Ali bin Abu Talib

The Companions unanimously agreed on the entitlement of ‘Ali ibn Abi Taalib (radiyallaahu ‘anhu) to the caliphate, and none among them contested this.

Therefore, whoever among them was present in Madinah after ‘Uthmaan’s murder (radiyallaahu ‘anhu) pledged their allegiance to him. The fact that those who were not in Madinah did not give their oath to him does not, in any way, imply that they disagreed with his appointment as caliph – just as the absence of some Companions during Abu Bakr’s (radiyallaahu ‘anhu) swearing-in did not imply their displeasure with his appointment.

As for the disputes which happened later on between the two camps of the Companions, it was because of the issue of handing over Uthmaan’s murderers for retribution, not because of any objection to ‘Ali’s right to the caliphate.

For more details, see:

Ali ibn Abi Talib (1/318-327), by Dr. ‘Ali M. Sallaabi.

And Allaah knows best.