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In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the five color theorem and generalized the four color conjecture to surfaces of arbitrary genus. [13] Tait, in 1880, showed that the four color theorem is equivalent to the statement that a certain type of graph (called a snark in modern terminology) must be non-planar. [14]
The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle ...
The discharging method is a technique used to prove lemmas in structural graph theory. [1] Discharging is most well known for its central role in the proof of the four color theorem. The discharging method is used to prove that every graph in a certain class contains some subgraph from a specified list.
Typically, the set S has four elements (the four colours of the four colour theorem), and c is a proper colouring, that is, each pair of adjacent vertices in V are assigned distinct colours. With these additional conditions, a and b are two out of the four colours available, and every element of the ( a , b )-Kempe chain has neighbours in the ...
A map of the United States using colors to show political divisions using the four color theorem. The first results about graph coloring deal almost exclusively with planar graphs in the form of map coloring. While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecture, noting
The four-color theorem states that every finite graph that can be drawn without crossings in the Euclidean plane needs at most four colors; however, some graphs with more complicated connectivity require more than four colors. [2]
The four color theorem was not given a valid proof until 1976. Kempe's proof can be translated into an algorithm to color planar graphs, which is also erroneous. Counterexamples to his proof were found in 1890 and 1896 (the Poussin graph), and later, the Fritsch graph and Soifer graph provided two smaller counterexamples. [3]
Robertson, Seymour & Thomas (1993) proved the conjecture for =, also using the four color theorem; their paper with this proof won the 1994 Fulkerson Prize. It follows from their proof that linklessly embeddable graphs, a three-dimensional analogue of planar graphs, have chromatic number at most five. [3]