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Assume that is disconnected, that is there exist two vertices, and , such that there is no path in between them. Let be the set of vertices incident to the vertex , and be the set of vertices incident to the vertex . Since and are not incident, then and .
If then all the sets are disjoint.
Hence, , or equivalently, , and , that is a contradiction to the condition .
From this it follows that the assumption is false, i.e. there exists a vertex . Thus, is incident to both the vertices and , and is a path in , connecting and , that is a contradiction with assumption that is disconnected.
Therefore, is a connected graph.