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The De Bruijn–Erdős theorem also applies directly to hypergraph coloring problems, where one requires that each hyperedge have vertices of more than one color. As for graphs, a hypergraph has a k {\displaystyle k} -coloring if and only if each of its finite sub-hypergraphs has a k {\displaystyle k} -coloring. [ 20 ]
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
A list edge-coloring is a choice of a color for each edge, from its list of allowed colors; a coloring is proper if no two adjacent edges receive the same color. A graph G is k-edge-choosable if every instance of list edge-coloring that has G as its underlying graph and that provides at least k allowed colors for each edge of G has
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.
The vertex coloring game was introduced in 1981 by Steven Brams as a map-coloring game [1] [2] and rediscovered ten years after by Bodlaender. [3] Its rules are as follows: Alice and Bob color the vertices of a graph G with a set k of colors. Alice and Bob take turns, coloring properly an uncolored vertex (in the standard version, Alice begins).
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.
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 ...
According to Jensen & Toft (1995), the problem was first formulated by Nelson in 1950, and first published by Gardner (1960). Hadwiger (1945) had earlier published a related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets, and he also mentioned the problem in a later paper (Hadwiger 1961).