Search results
Results from the WOW.Com Content Network
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.
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.
For a graph class G, the defective chromatic number of G is minimum integer k such that for some integer d, every graph in G is (k,d)-colourable. For example, the defective chromatic number of the class of planar graphs equals 3, since every planar graph is (3,2)-colourable and for every integer d there is a planar graph that is not (2,d ...
The Fraysseix–Rosenstiehl planarity criterion can be used directly as part of algorithms for planarity testing, while Kuratowski's and Wagner's theorems have indirect applications: if an algorithm can find a copy of K 5 or K 3,3 within a given graph, it can be sure that the input graph is not planar and return without additional computation.
However, there exist fast algorithms for this problem: for a graph with n vertices, it is possible to determine in time O(n) (linear time) whether the graph may be planar or not (see planarity testing). For a simple, connected, planar graph with v vertices and e edges and f faces, the following simple conditions hold for v ≥ 3: Theorem 1. e ...
By applying exact algorithms for vertex coloring to the line graph of the input graph, it is possible to optimally edge-color any graph with m edges, regardless of the number of colors needed, in time 2 m m O(1) and exponential space, or in time O(2.2461 m) and only polynomial space (Björklund, Husfeldt & Koivisto 2009).
George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem.If (,) denotes the number of proper colorings of G with k colors then one could establish the four color theorem by showing (,) > for all planar graphs G.