Some basic Theorems about Graphs in Exercises: Part 17.

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This story belongs to a contiguous series with the aim to list some of the basic theorems of graph theory. It is intentionally written in form of exercises to invite the reader having a ‘learning experience by doing’. In this sense it is not important to have solved any of these exercises but it is important to have tried them for a couple of minutes.

As usually in this series, all graphs are assumed to be simple, undirected and finite, except otherwise mentioned.

Problem 1:

A graph G that fulfills deg(v) + deg(w) >/= ord(G) for all distinct non-adjacent vertexes v, w, is connected.

Remember ord(G) is the number of vertexes and deg(x) the number of all adjacent vertexes of x.

Solution:

Assume to the contrary that G is not connected. Then we can split G into the graphs H and R such that no vertex of H can be connected to a vertex in R. On the one hand we have

ord(G) = ord(H) + ord(R)