Show that a 6regular graph on 10 vertices cannot be bipartit

Show that a 6-regular graph on 10 vertices cannot be bipartite.

Solution

A 6 - regular graph with 10 vertices has 6x10/2 = 30 edges.

A bipartite graph with 10 vertices has atmost 10²/4 = 25 edges

clearly the maximum number of edges of a bipartite graph with 10 vertices can not be equal to the 6 - regular graph with 10 vertices.

Hence , a 6 - regular graph with 10 vertices can not be a bipartite