In Jarvan IV's Relations, it said this quote.
Jarvan IV is the arch-enemy of Jericho Swain.
I looked at their lore.
And didn't find any information about this relationship.
Why they are arch-enemies? When did this occur?
Try reading the first paragraph of Jarvan's lore again; I'll quote the relevant section:
Jarvan hates Swain because Swain captured him and nearly had Jarvan executed.
Additionally, Swain hates Jarvan because Noxus hates Demacia, and the two are each prominent leaders of their respective city-states.
If someone tried to have you executed after embarrassing you and slaughtering all your men, wouldn't you hold a grudge? ;)
From Swain's League Judgement:
Not to mention all the fighting over Kalamanda, which can be read about in the Journal of Justice, especially issues 11, 18, 20, 21, 23, 24, 25, 27, and 29. In issue 25, Jarvan and Swain get into a fist-fight as war seems inevitable.