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Graph coloring is computationally hard. It is NP-complete to decide if a given graph admits a k-coloring for a given k except for the cases k ∈ {0,1,2}. In particular, it is NP-hard to compute the chromatic number. [33] The 3-coloring problem remains NP-complete even on 4-regular planar graphs. [34]
Graph coloring [2] [3]: GT4 Graph homomorphism problem [3]: GT52 Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph.
Print/export Download as PDF ... Pages in category "NP-hard problems" The following 20 pages are in this category, out of 20 total. ... Radio coloring;
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).
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In graphs where every vertex has at most three neighbors, the clique cover remains NP-hard, and there is a constant ρ > 1 such that it is NP-hard to approximate with approximation ratio ρ or better. Nevertheless, in polynomial time it is possible to find an approximation with a ratio of 5/4. That is, this approximation algorithm finds a ...
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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.