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For each surface $$\Sigma $$ Σ , we define $$\Delta (\Sigma ) =$$ Δ ( Σ ) = max $$\{\Delta (G)|\,G$$ { Δ ( G ) | G is a class two graph with maximum degree $$\Delta (G)$$ Δ ( G ) that can be embedded in $$\Sigma \}$$ Σ } . Hence Vizing’s Planar Graph Conjecture can be restated as $$\Delta (\Sigma )=5$$ Δ ( Σ ) = 5 if $$\Sigma $$ Σ...
When the vertices and edges are colored with k colors, an edge is called monochromatic if the edge and the two vertices incident with it all have the same color. The adapted chromatic number of a graph G, $$\chi _{a}\left( G\right) ,$$ χ a G , is the least integer k such that for each k-edge coloring of G the vertices of G can be colored with the same set of colors without creating...
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