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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.
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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.
A proper distinguishing coloring is a distinguishing coloring that is also a proper coloring: each two adjacent vertices have different colors. The minimum number of colors in a proper distinguishing coloring of a graph is called the distinguishing chromatic number of the graph. [12]
It indicates how to give color to geographic areas (common geopolitical delimitations: nations, regions, etc.). With the following steps: Choose the colors to paint the areas. Choose for one of two possibilities: Paint the areas of a blank map. Indicate that areas are still painted (only for maps of the world).
An extension of Brooks' theorem to total coloring, stating that the total chromatic number is at most Δ + 2, has been conjectured by Mehdi Behzad and Vizing. The Hajnal–Szemerédi theorem on equitable coloring states that any graph has a (Δ + 1)-coloring in which the sizes of any two color classes differ by at most one.
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