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The empire problem, also known as the m-pire problem) asks for the maximum number of colors needed to color countries such that no two countries sharing a common border have ...

The bound for the number of colors which are sufficient for map coloring on a surface of genus g, gamma(g)=|_1/2(7+sqrt(48g+1))_| is the best possible, where |_x_| is the ...

Given a map with genus g>0, Heawood showed in 1890 that the maximum number N_u of colors necessary to color a map (the chromatic number) on an unbounded surface is N_u = ...

Connect-Four is a tic-tac-toe-like two-player game in which players alternately place pieces on a vertical board 7 columns across and 6 rows high. Each player uses pieces of ...

The Franklin graph is the 12-vertex cubic graph shown above whose embedding on the Klein bottle divides it into regions having a minimal coloring using six colors, thus ...

Tait's Hamiltonian graph conjecture asserted that every cubic polyhedral graph is Hamiltonian. It was proposed by Tait in 1880 and refuted by Tutte (1946) with a ...

Let G be a planar graph whose vertices have been properly colored and suppose v in V(G) is colored C_1. Define the C_1C_2-Kempe chain containing v to be the maximal connected ...

Assignment of each graph edge of a graph to one of two color classes (commonly designation "red" and "green").

Let a tree S be a subgraph of a cubic graph G. The graph excision G circleminus S is the graph resulting from removing the tree, then merging the edges. For example, if in ...

The Hadwiger conjecture is a generalization of the four-color theorem which states that for any loopless graph G with h(G) the Hadwiger number and chi(G) the chromatic ...