Edges, Vertices and Faces in Hex 

Contexts in source publication

Context 1

... Nacional de Colombia Sede Bogotá -Facultad de Ciencias Económicas the demonstration of the Hex theorem, we have to think of the Hex board as a graph 10 , in order to do that it is necessary to know the very basic concepts of graph theory edge, vertex and face. A vertex in the Hex board would look like the blue point in figure 6a; we can understand it as any corner of any hex, so in the figure 6a there are 13 vertices. In the same figure we can also see three edges in red; so an edge will be any border of any hex, thus in the figure there are a total of 15 edges. ...

Context 2

... under this construction (í µí°» ̂ ) will have three types of degree; 0, 1 and 2. It is impossible to have a vertex with a greater degree than 2 or for one to have a degree of 1 while not being in any corner. In a Graph made by adjoint hexes, as the Hex board (leaving no free spaces in it, as the Hex board does), í µí°º = (í µí±‰′, í µí°¸′), with V all the vertices and E all the edges (unlike í µí°» ̂ , so If í µí°º and í µí°» ̂ are in the same hex board then |V ′ | ≥ |V|, it is a broader definition) the maximum degree of any vertex in V will be 3. Now we apply the conditions of í µí±‰ to those vertices, that means that the maximum degree is reduced to 2. Let's take a vertex í µí±‰ 2 ∈ í µí±‰ in í µí°» ̂ , with í µí±(í µí±‰ 2 ) = 3, therefore it has 3 incident edges í µí±’ í µí±– í µí¼– E with í µí±– = 1, 2, 3, it would look like the blue point in figure 6a; given that the edge í µí°µ is in í µí°¸thereforeµí°¸therefore í µí°´′ and í µí°µ′ must be different colors, and given that the í µí° ¶ edge is in í µí°¸, then í µí° ¶′ and í µí°µ′ are opposite colors, therefore if í µí°´′ and í µí° ¶′ are opposite colors from í µí°µ′, then they are the same color, ...