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In graph-theoretic terms, the theorem states that for a 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 graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph, illustrating the case k = 4 of Hadwiger's conjecture In graph theory , the Hadwiger conjecture states that if G {\displaystyle G} is loopless and has no K t {\displaystyle K_{t}} minor then its chromatic number satisfies ...
GCol An open-source python library for graph coloring. High-Performance Graph Colouring Algorithms Suite of 8 different algorithms (implemented in C++) used in the book A Guide to Graph Colouring: Algorithms and Applications (Springer International Publishers, 2015). CoLoRaTiOn by Jim Andrews and Mike Fellows is a graph coloring puzzle
This would have established the four color theorem. No graph can be 0-colored, so 0 is always a chromatic root. Only edgeless graphs can be 1-colored, so 1 is a chromatic root of every graph with at least one edge. On the other hand, except for these two points, no graph can have a chromatic root at a real number smaller than or equal to 32/27. [8]
The discharging method is a technique used to prove lemmas in structural graph theory. [1] Discharging is most well known for its central role in the proof of the four color theorem. The discharging method is used to prove that every graph in a certain class contains some subgraph from a specified list.
Finding ψ(G) is an optimization problem.The decision problem for complete coloring can be phrased as: . INSTANCE: a graph G = (V, E) and positive integer k QUESTION: does there exist a partition of V into k or more disjoint sets V 1, V 2, …, V k such that each V i is an independent set for G and such that for each pair of distinct sets V i, V j, V i ∪ V j is not an independent set.
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.
Suppose again that G is a graph, with edge set E, and this time we have a colouring function c : E → S . {\displaystyle c:E\to S.} If e is an edge assigned colour a , then the ( a , b )-Kempe chain of G containing e is the maximal connected subset of E which contains e and whose edges are all coloured either a or b .