The problem is : There used to be 26 teams with 13 teams in each of two conferences. Each team’s 14-game schedule should include exactly 11 games against teams in its own conference and three games against teams in the other conference. Show that this guideline could not be satisfied!
My thought : Firstly there will be total of $\frac{(26*3)}{2} = 39$ matches where a team plays with another team of different conference. Such 39 matches have to be arranged in 14 time slots such that each team plays in 3 matches with a team of opposite conference, i.e each team require 3 time slots.
So I think the problem is reduced to if there exists a graph with 39 edges and where each vertex has degree 3 and length of largest non-circuit is 14.
Firstly am I thinking right? I don't have much grasp in graph theory and don't have clue how to solve it, except maybe but some sort of brute force. Also if this translates to some sort of coloring problem or anything else, I am hopelessly lost. (I only know the four color theorem statement which says every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color)
The graph they are talking about has one vertex for each team and an edge between two vertices if the two teams play one of these games. The graph is bipartite with the teams in one conference forming one vertex set.
Your computation of $39$ games corresponds to there being $39$ edges in this graph. By considering the games within a conference, you should be able to show that there must be an odd number of interconference games each week. Since there is an even number of weeks.....
I don't think the graph adds anything to the problem. You can follow the logic in the previous paragraph without the graph.