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In graph-theoretic terms, the theorem states that for a loopless planar graph, its chromatic number is ().. The intuitive statement of the four color theorem – "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color" – needs to be interpreted appropriately to be correct.
The four pigments in a bird's cone cells (in this example, estrildid finches) extend the range of color vision into the ultraviolet. [1]Tetrachromacy (from Greek tetra, meaning "four" and chroma, meaning "color") is the condition of possessing four independent channels for conveying color information, or possessing four types of cone cell in the eye.
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. The presence of the desired subgraph is then often used to prove a coloring result. [1]
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 graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph, illustrating the case k = 4 of Hadwiger's conjecture In graph theory , the Hadwiger conjecture states that if G {\displaystyle G} is loopless and has no K t {\displaystyle K_{t}} minor then its chromatic number satisfies ...
With four colors, it can be colored in 24 + 4 × 12 = 72 ways: using all four colors, there are 4! = 24 valid colorings (every assignment of four colors to any 4-vertex graph is a proper coloring); and for every choice of three of the four colors, there are 12 valid 3-colorings. So, for the graph in the example, a table of the number of valid ...
Chromatic roots have been very well studied, in fact, Birkhoff’s original motivation for defining the chromatic polynomial was to show that for planar graphs, (,) > for x ≥ 4. This would have established the four color theorem. No graph can be 0-colored, so 0 is always a chromatic root.
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 ...