I need to show that if G is a planar graph, then G must have a vertex of degree at most 5.
A graph is called a planar graph if it is drawn in such a way that the edges never cross, except at where they meet at vertices.
Let G be a planar graph with k connected components, n vertices, q edges, and r regions, we have to prove, that the average degree of the vertices is less than 6.
We have from graph theory,
deg(v1) + deg(v2) + deg(v3) + ... + deg(vn) = 2q
where v1, v2, v3, ... vn denote the vertices in G
Average degree of the vertices is A = (1/n)*2q .....(1)
but we have from another theorem, q ≤ 3n − 6 ...
This is a proof regarding the degree of a planar graph.