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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.
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.
Acyclic coloring. Every graph with acyclic chromatic number has () (+). [7]Marking game. For every graph , () (), where () is the game coloring number of .Almost every known upper bound for the game chromatic number of graphs are obtained from bounds on the game coloring number.
The total chromatic number χ″(G) of a graph G is the fewest colors needed in any total coloring of G. The total graph T = T(G) of a graph G is a graph such that (i) the vertex set of T corresponds to the vertices and edges of G and (ii) two vertices are adjacent in T if and only if their corresponding elements are either adjacent or incident ...
The minimum number of colors needed for the incidence coloring of a graph G is known as the incidence chromatic number or incidence coloring number of G, represented by (). This notation was introduced by Jennifer J. Quinn Massey and Richard A. Brualdi in 1993. Incidence coloring of a Petersen graph
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
In graph theory, circular coloring is a kind of coloring that may be viewed as a refinement of the usual graph coloring. The circular chromatic number of a graph G {\displaystyle G} , denoted χ c ( G ) {\displaystyle \chi _{c}(G)} can be given by any of the following definitions, all of which are equivalent (for finite graphs).
The chromatic number of a graph is the minimum number of colors in any coloring. The colorings shown are optimal, so the chromatic number is three for the 7-cycle and four for the other graph shown. The vertices of any clique must have different colors, so the chromatic number is always greater than or equal to the clique number.