Combinatorics and Graph Theory

A graph \(G\) consists of a vertex set \(V\) and a collection \(E\) of \(2\)-element subsets of \(V\). Elements of \(E\) are called edges. In our course, we will (almost always) use the convention that \(V=\{1,2,3,\dots,n\}\) for some positive integer \(n\). With this convention, graphs can be described precisely with a text file:

1. The first line of the file contains a single integer \(n\), the number of vertices in the graph.

2. Each of the remaining lines of the file contains a pair of distinct integers and specifies an edge of the graph.

We illustrate this convention in with a text file and the diagram for the graph\(G\) it defines.

Much of the notation and terminology for graphs is quite natural. See if you can make sense out of the following statements which apply to the graph \(G\) defined above:

1. \(G\) has \(9\) vertices and \(10\) edges.

2. \(\{2,6\}\) is an edge.

3. Vertices \(5\) and \(9\) are adjacent.

4. \(\{5,4\}\) is not an edge.

5. Vertices \(3\) and \(7\) are not adjacent.

6. \(P = (4, 3,1, 7,9,5)\) is a path of length\(5\) from vertex \(4\) to vertex\(5\).

7. \(C=(5,9,7,1)\) is cycle of length\(4\).

8. \(G\) is disconnected and has two components. One of the components has vertex set \(\{2,6,8\}\).

9. \(\{1,5,7\}\) is a triangle.

10. \(\{1,7,5,9\}\) is a clique of size\(4\).

11. \(\{4,2,8,5\}\) is an independent set of size\(4\).

Equipped only with this little bit of background material, we are already able to pose a number of interesting and challenging problems.

We will frequently study problems in which graphs arise in a very natural manner. Here's an example.

Condensed: the full section is in Keller & Trotter, Applied Combinatorics.