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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.
Poh [3] and Goddard [4] showed that any planar graph has a special (3,2)-coloring in which each color class is a linear forest, and this can be obtained from a more general result of Woodall. For general surfaces, it was shown that for each genus g ≥ 0 {\displaystyle g\geq 0} , there exists a k = k ( g ) {\displaystyle k=k(g)} such that every ...
The Recursive Largest First (RLF) algorithm is a heuristic for the NP-hard graph coloring problem. It was originally proposed by Frank Leighton in 1979. [1] The RLF algorithm assigns colors to a graph’s vertices by constructing each color class one at a time.
A particular case is L(2,1)-coloring. Oriented coloring Takes into account orientation of edges of the graph Path coloring Models a routing problem in graphs Radio coloring Sum of the distance between the vertices and the difference of their colors is greater than k + 1, where k is a positive integer. Rank coloring
Given a graph G and given a set L(v) of colors for each vertex v (called a list), a list coloring is a choice function that maps every vertex v to a color in the list L(v). As with graph coloring, a list coloring is generally assumed to be proper , meaning no two adjacent vertices receive the same color.
In graph theory, the Erdős–Faber–Lovász conjecture is a problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972. [1] It says: If k complete graphs , each having exactly k vertices, have the property that every pair of complete graphs has at most one shared vertex, then the union ...
A frequent goal in graph coloring is to minimize the total number of colors that are used; the chromatic number of a graph is this minimum number of colors. [1] 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 ...
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring [1] is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but ...