Graph Theory 4

Complete graph. A complete graph is an undirected graph with no multiple edges and no loops whereeach vertex is adjacent to (has an edge to) every other vertex. Kn is the complete graph on n vertices. Belowis the graph K4.

Note edges AC and BD appear to intersect. But they do not. With graphs, edges only intersect at vertices.Thus the apparent intersection of AC and BD is the fault of the drawing. If you want, you can think thatBD dips down a little and is actually below AC at apparent crossing.

For the defintion of Eulerian circuits and trails below, we allow a path to visit a vertex more than once.

Euler Circuit. An Euler circuit of a graph is a path that begins and ends at the same vertex and goesthrough each edge exactly once. Example:

An Euler Circuit for the graph above is [A,C,D,E,C,B,A].

1. Draw K3.

2. Draw K5.

3. What is the degree of a vertex in Kn? Recall the degree of a vertex is the number of edges that touch thevertex.

4. An Euler trail is like an Euler circuit except it does NOT have to start and end at the same vertex. Itgoes each edge of the graph exactly once. Find an Euler trail for the graph below: