User:Wysici

An independent vertex set (IVS) in graph G is a clique in GC, the complement of G. Then every clique in GC is congruent to an independent set of vertices in G; all of which, may have the same color. Lemma: If GC can be partitioned into exactly 4 independent maximal cliques, then G is 4-chroma. Four cliques is congruent to four colors. It will be shown how this works by analyzing graphs of small order, N = 5 thru 10

Lemma: If G is Hamiltonian, it may be drawn as an N-sided polygon; with (N-3) inner diagonals and (N-3) external diagonals. A maximal planar graph has 3*N-6 edges. An MPG drawn this way has

N + (N-3) + (N-3) = 3*N-6 edges

Let Eg be the number of edges in G. then, Eg = 3*N-6 Let Ec be the number of edges in a complete graph. Then Ec = (N^2-N)/2. Let Ex be the number of edges in the complement of a MPG. Then Ex = Ec-Eg = (N^2-N)/2 - 3*N+6 = N^2 - 7*N + 12 = [(N-3)*(N-4)]/2

Let G be an MPG of order 5. Then Ex = (2*1)/2 = 1.

One edge in the complement shows that 2 vertices in the MPG are not adjacent. Therefore, a MPG of order 5 is always 4-chroma.

Wysici (talk) 00:30, 21 July 2024 (UTC)