Graph Theory

Below are a series of problems that I have been given in my Introduction to Graph Theory class this semester at SJSU. Please take a moment to walk through the proofs. They are as much my attempt to understand them as they are interesting and fun to parse.

= Homework Set #1 = There used to be 26 teams in the NFL, with 13 teams in each of the two conferences (AFC and NFC). An NFL Guidebook said each team is to play a 14 game schedule, consisting of 11 games within its own conference, and 3 against teams in the other conference. Is this possible?

The problem with this situation is subtle, but requires noting a couple key notes. Considering this is a study of graph theory we have to keep that in mind before we go too far. This is a study of graphs and how they interconnect. This question is actually a compound problem.

Is it possible for an odd number of vertices to have an odd degree? The induced subgraph of either the NFC or AFC is 13 nodes in size. Is it possible that this subgraph can connect with 11 other nodes and only 11 other nodes? Since a graph is composed of only two elements Nodes and Edges, and an edge can only exist if it is between two vertices. A graph of two vertices and an edge symbolizes a single game between two teams.

The proof is not difficult, but i will approach it at a later time. Suffice to say that this requirement is the limiting factor in our graph.