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The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f(v) to each vertex v, a graph G is f-choosable (or f-list-colorable) if it has a list coloring no matter how one assigns a list of f(v) colors to ...
The smallest number of colors needed for an edge coloring of a graph G is the chromatic index, or edge chromatic number, χ ′ (G). A Tait coloring is a 3-edge coloring of a cubic graph. The four color theorem is equivalent to the assertion that every planar cubic bridgeless graph admits a Tait coloring.
The concept was introduced by William K. Hale. [2] If T = {0} it reduces to common vertex coloring. The T-chromatic number, (), is the minimum number of colors that can be used in a T-coloring of G. The complementary coloring of T-coloring c, denoted ¯ is defined for each vertex v of G by
The user selects the color corresponding to one of the numbers then uses it to fill in a delineated section of the canvas, in a manner similar to a coloring book. The kits were invented, developed and marketed in 1950 by Max S. Klein, an engineer and owner of the Palmer Paint Company in Detroit, Michigan, United States, and Dan Robbins, a ...
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory.Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.
Finding ψ(G) is an optimization problem.The decision problem for complete coloring can be phrased as: . INSTANCE: a graph G = (V, E) and positive integer k QUESTION: does there exist a partition of V into k or more disjoint sets V 1, V 2, …, V k such that each V i is an independent set for G and such that for each pair of distinct sets V i, V j, V i ∪ V j is not an independent set.
A Grundy coloring of a t-atom can be obtained by coloring the independent set first with the smallest-numbered color, and then coloring the remaining (t − 1)-atom with an additional t − 1 colors. For instance, the only 1-atom is a single vertex, and the only 2-atom is a single edge, but there are two possible 3-atoms: a triangle and a four ...
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.