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42 votes

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 $G_1 \cap G_2= (V,E_1\cap E_2)$ is not a connected graph, then the graph $G_1\cup G_2=(V,E_1\cup E_2)$

- 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 $G_2$

9

Refere http://math.stackexchange.com/questions/1535029/union-of-two-graphs This question, if you are not able to interpret this image.

0 votes

Since there are two connected graphs G1 and G2 for the same set of vertices.

Now, __G1 intersection G2 is disconnected means G1 and G2 must have at least one different path between two vertices__.

So, for that pair of vertex cycle is formed