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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 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 graph coloring associates each vertex with a color drawn from a set of colors, in such a way that every edge has two different colors at its endpoints. 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]
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 degree of a graph also appears in upper bounds for other types of coloring; for edge coloring, the result that the chromatic index is at most Δ + 1 is Vizing's theorem. 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 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
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 chromatic number of the flower snark J 5 is 3, but the circular chromatic number is ≤ 5/2.. In graph theory, circular coloring is a kind of coloring that may be viewed as a refinement of the usual graph coloring.
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