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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 ...
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring [1] is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but ...
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 adjacent-vertex-distinguishing-total-chromatic number χ at (G) of a graph G is the fewest colors needed in an AVD-total-coloring of G. The following lower bound for the AVD-total chromatic number can be obtained from the definition of AVD-total-coloring: If a simple graph G has two adjacent vertices of maximum degree, then χ at ( G ) ≥ ...
Sulanke's nine-color Earth–Moon map (left and right), with adjacencies described by the join of a 6-vertex complete graph and 5-vertex cycle graph (center) An example of a biplanar graph requiring 9 colors can be constructed as the join of a 6-vertex complete graph and a 5-vertex cycle graph. This means that these two subgraphs are connected ...
Graph is vertex-critical if and only if for every vertex , there is an optimal proper coloring in which is a singleton color class. As Hajós (1961) showed, every k {\displaystyle k} -critical graph may be formed from a complete graph K k {\displaystyle K_{k}} by combining the Hajós construction with an operation that identifies two non ...