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The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f ( v ) to each vertex v , a graph G is f -choosable (or f -list-colorable ) if it has a list coloring no matter how one assigns a list of f ( v ...
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 graph of the 3-3 duoprism (the line graph of ,) is perfect.Here it is colored with three colors, with one of its 3-vertex maximum cliques highlighted. In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph.
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 ...
A frequent goal in graph coloring is to minimize the total number of colors that are used; the chromatic number of a graph is this minimum number of colors. [1] The four-color theorem states that every finite graph that can be drawn without crossings in the Euclidean plane needs at most four colors; however, some graphs with more complicated ...
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 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.
Let K(H) is the chromatic number of any ordered graph H. Then for any ordered graph H, X < (H) ≥ K(H). One thing to be noted, for a particular graph H and its isomorphic graphs the chromatic number is same, but the interval chromatic number may differ. Actually it depends upon the ordering of the vertex set.