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For planar graphs, vertex colorings are essentially dual to nowhere-zero flows. About infinite graphs, much less is known. The following are two of the few results about infinite graph coloring: If all finite subgraphs of an infinite graph G are k-colorable, then so is G, under the assumption of the axiom of choice.
In graph-theoretic terms, the theorem states that for loopless planar graph, its chromatic number is ().. The intuitive statement of the four color theorem – "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color" – needs to be interpreted appropriately to be correct.
A 3-coloring of a triangle-free planar graph. In the mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors.
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.
Expressed more formally, this reasoning implies that if a graph has m edges in total, and if at most β edges may belong to a maximum matching, then every edge coloring of the graph must use at least m/β different colors. [4] For instance, the 16-vertex planar graph shown in the illustration has m = 24 edges. In this graph, there can be no ...
We will use a representation of the graph in which each vertex maintains a circular linked list of adjacent vertices, in clockwise planar order. In concept, the algorithm is recursive, reducing the graph to a smaller graph with one less vertex, five-coloring that graph, and then using that coloring to determine a coloring for the larger graph ...
Misra & Gries (1992) describe a polynomial time algorithm for coloring the edges of any graph with Δ + 1 colors, where Δ is the maximum degree of the graph. That is, the algorithm uses the optimal number of colors for graphs of class two, and uses at most one more color than necessary for all graphs.
A planar graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph K 2,4). A sufficient condition that a graph can be drawn convexly is that it is a subdivision of a 3-vertex-connected planar graph.