P(G - uv, k) represents the number of possible proper colorings of the graph, when the vertices may have same or different colors.

If the algorithm terminates without finding an odd cycle in this way, then it must have found a proper coloring, and can safely conclude that the graph is bipartite.

Just as no graph with a loop edge has a proper coloring, no graph with a bridge can have a nowhere-zero flow (in any group).

If denotes the number of proper colorings of with colors then one could establish the four color theorem by showing for all planar graphs .

Then is a proper coloring of if and only if, for all there exists such that .

The number of proper colorings therefore come from the sum of two graphs.

Somewhat surprisingly, efficient decentralized algorithms exist that will color a graph if a proper coloring exists.

A complete graph is uniquely colorable, because the only proper coloring is one that assigns each vertex a different color.

The graphs formed in this way always require k colors in any proper coloring.

We went to a Polynesian bar, where my attorney made seventeen calls before locating a convertible with adequate horsepower and proper coloring.