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Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring ...
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.
Exact coloring of the complete graph K 6. Every n-vertex complete graph K n has an exact coloring with n colors, obtained by giving each vertex a distinct color. Every graph with an n-color exact coloring may be obtained as a detachment of a complete graph, a graph obtained from the complete graph by splitting each vertex into an independent set and reconnecting each edge incident to the ...
An example of defective colouring of a cycle on five vertices, , is as shown in the figure. The first subfigure is an example of proper vertex colouring or a (k, 0)-coloring. The second subfigure is an example of a (k, 1)-coloring and the third subfigure is an example of a (k, 2)-coloring. Note that,
A more general version of the theorem applies to list coloring: given any connected undirected graph with maximum degree Δ that is neither a clique nor an odd cycle, and a list of Δ colors for each vertex, it is possible to choose a color for each vertex from its list so that no two adjacent vertices have the same color. In other words, the ...
The star K 1,5 - a single central vertex connected to five others - is a complete bipartite graph, and therefore may be colored with two colors. However, the resulting coloring has one vertex in one color class and five in another, and is therefore not equitable.
For a given graph, the minimum span over all possible labelling functions is the λ h,k-number of G, denoted by λ h,k (G). When h = 1 and k = 0, it is the usual (proper) vertex coloring. There is a very large number of articles concerning L(h, k)-labelling, with different h and k parameters and different classes of graphs.
A graph vertex coloring is a weak coloring, but not necessarily vice versa. Every graph has a weak 2-coloring. The figure on the right illustrates a simple algorithm for constructing a weak 2-coloring in an arbitrary graph. Part (a) shows the original graph. Part (b) shows a breadth-first search tree of the same graph. Part (c) shows how to ...