Congruence Of Two Vertices Lab Report

Assume the CAVA confirms the existence of an unavoidable conflict. The congruent coloring requires the fixed color be chosen, and consequently the two top rims be congruent conflict free. The focus is on these two issues.
Congruence of Two Vertices
Within three colors, two vertices (p, q)V(G) can be in one of the congruence relation defined as: congruent c(p) ≅ c(q), the resulting color c(q) always equals the selected color for c(p). anti-congruent c(p) c(q), <p, q>E, the resulting color c(q) always differs from the selected color for p. See figure 9 for an example of anti-congruence.
The third alternative when the colors of vertices are independent of each other is not used.
Let circuit Cf be defined: Cf = ∪(∆(E(C)) (CNf), and graph
Proof. A G1L graph 3-coloring may not be feasible, regardless of the sub-rim vertices colors, when exist two connected congruent vertices. Sub-graph G1L received from F[5], denoted by G1F, is such and the only
Let investigate graph G2TR under what circumstances its derived G1L will have a congruence conflict. A cycle Cf' that exhibits conflict is a C5 cycle with vertex v congruent to connected vertices p, q either within rim Ry or Rx. The cycle Cf' cannot be of an even length or C7+ cycle; the congruent vertices are not connected. More numerous presence of {f*} within P2R prevents from congruence of vertices within cycles {Cf} where fPy.
It is also impossible to have connected congruent vertices pRx and qRy since vertex p has no associated triangles to be congruent with another vertex. That means vertices s, t adjacent to p, q respectively would be within the same rim Ry as color reduced vertex f. However in order to have triangles to carry the congruence, vertices s, t should not be included into rim Ry since they would have also CF color. Thus vertices should be within rim Rx. This is a clear contradiction of the requirements for vertices s, t.
That contradiction ceases to be a contradiction only if Cf' is exactly as Cf* within F. Vertices f, p, and q are required to form a triangle of diagonals to carry the congruence via edge adjacent