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  "body": "\n## Summary\n\nA [[graph coloring|0NCPXH194Y]] is a labeling of the vertices of a graph by elements from a given set of colors, or equivalently a partition of the vertices into subsets, called \"color classes\", each of which is associated with one of the colors.\n\nSome authors use \"coloring\", without qualification, to mean a proper coloring, one that assigns different colors to the endpoints of each edge. In graph coloring, the goal is to find a proper coloring that uses as few colors as possible; for instance, [[bipartite graphs|07DVHJ8N0A]] are the graphs that have colorings with only two colors, and the [[four color theorem|0M406JC8J1]] states that every [[planar graph|029JQEK15B]] can be colored with at most four colors. A graph is said to be $k$-colored if it has been (properly) colored with $k$ colors, and $k$-colorable or $k$-chromatic if this is possible.\n\nMany variations of coloring have been studied, including [[edge coloring|0B5WCC02AN]] (coloring edges so that no two edges with the same endpoint share a color), list coloring (proper coloring with each vertex restricted to a subset of the available colors), acyclic coloring (every 2-colored subgraph is acyclic), co-coloring (every color class induces an independent set or a clique), complete coloring (every two color classes share an edge), and total coloring (both edges and vertices are colored).\n\nThe coloring number of a graph is one plus the degeneracy. It is so called because applying a greedy coloring algorithm to a degeneracy ordering of the graph uses at most this many colors.\n",
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