$1)$ A signed graph $G$ is N-balanced if every circuit of length at most N is positive. Give an example of a signed graph which is 3-balanced but not 4 balanced?
I tried drawing a square and made it 3 balance and not 4 balanced but what throws me off is that it says circuit at most N is positive ? does that affect the number of positive signs i must have?
$2)$ A signed graph is locally balanced at vertex u if every circuit containing u is positive. Show that a signed graph may be locally balanced at some vertex u without being (globally) balanced.
I'm not to sure about my answer but i just want someone to check it but idk how to post it since it hand drawn.
Compare to my examples ;)
A square with edge weight -1 is 3 balanced (it has no 3 circuits, so no-non-positive ones). It is not 4-balanced
Two separate triangles, one with all edges -1 and one with all edges +1. Any $u$ on the second one is locally balanced. Any $v$ on the first one is not locally balanced, to the whole graph is not locally balanced.