Generalization of the liar's paradox

 The system is a generalization of the liar’s paradox in graphs. Instead of there being one agent that can be a liar or a truthful agent speaking about itself (the liar’s paradox), there are n agents that could be either liars or truthful agents speaking about themselves and other agents.

Theorem 1: a generalized liar’s paradox system is consistent only if all cycles have an even number of falses.

Problem 1: Given an edge bicolored (red and black) graph, assess whether all cycles have an even number of black edges.

Problem 2: Given an edge bicolored (red and black) graph, count all the cycles that have an even number of black edges.