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The chromatic number of the flower snark J 5 is 3, but the circular chromatic number is ≤ 5/2. In graph theory, circular coloring is a kind of coloring that may be viewed as a refinement of the usual graph coloring. The circular chromatic number of a graph , denoted () can be given by any of the following definitions, all of which are ...
A proper distinguishing coloring is a distinguishing coloring that is also a proper coloring: each two adjacent vertices have different colors. The minimum number of colors in a proper distinguishing coloring of a graph is called the distinguishing chromatic number of the graph. [12]
For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. ch(G) ≥ χ(G).A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. The answer is unknown, but has been narrowed down to one of the numbers 5, 6 or 7.
George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem.If (,) denotes the number of proper colorings of G with k colors then one could establish the four color theorem by showing (,) > for all planar graphs G.
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 ...
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Above:A 3:1-coloring of the cycle on 5 vertices, and the corresponding 6:2-coloring. Below: A 5:2 coloring of the same graph. A b-fold coloring of a graph G is an assignment of sets of size b to vertices of a graph such that adjacent vertices receive disjoint sets. An a:b-coloring is a b-fold coloring out of a available colors.