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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.
In graph theory, Brooks' theorem states a relationship between the maximum degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertices can be colored with only Δ colors, except for two cases, complete graphs and cycle graphs of odd length, which require ...
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.
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem .
The 7 cycles of the wheel graph W 4. For odd values of n, W n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even n, W n has chromatic number 4, and (when n ≥ 6) is not perfect. W 7 is the only wheel graph that is a unit distance graph in the ...
The incidence game chromatic number of a graph G, denoted by (), is the fewest colors required for Alice to win in an incidence coloring game. It unifies the ideas of incidence chromatic number of a graph and game chromatic number in case of an undirected graph.
The edge choosability, or list edge colorability, list edge chromatic number, or list chromatic index, ch'(G) of graph G is the least number k such that G is k-edge-choosable. It is conjectured that it always equals the chromatic index.
Therefore, the chromatic number of a subgraph cannot be larger than the chromatic number of the whole graph. The De Bruijn–Erdős theorem concerns the chromatic numbers of infinite graphs, and shows that (again, assuming the axiom of choice) they can be calculated from the chromatic numbers of their finite subgraphs.