Let $G_1 = (V, E_1)$ and $G_2 = (V, E_2)$ be connected graphs on the same vertex set $V$ with more than two vertices. If $G_1 ∩ G_2 = (V, E_1 ∩ E_2)$ is not a connected graph, then the graph $G_1 U G_2 = (V, E_1 U E_2)$
- cannot have a cut vertex
- must have a cycle
- must have a cut-edge (bridge)
- has chromatic number strictly greater than those of $G_1$ and $G_2$
My attempt :
Somewhere, answer is given option $(2)$ , but it may be false , counter example is take line graph. It seems to option $(3)$ is true .
Can you explain please ?