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A similar theorem states that K 4 and K 2,3 are the forbidden minors for the set of outerplanar graphs. Although the Robertson–Seymour theorem extends these results to arbitrary minor-closed graph families, it is not a complete substitute for these results, because it does not provide an explicit description of the obstruction set for any family.
Robertson, Seymour & Thomas (1993) proved the conjecture for =, also using the four color theorem; their paper with this proof won the 1994 Fulkerson Prize. It follows from their proof that linklessly embeddable graphs , a three-dimensional analogue of planar graphs, have chromatic number at most five. [ 3 ]
As the Robertson–Seymour theorem shows, many important families of graphs can be characterized by a finite set of forbidden minors: for instance, according to Wagner's theorem, the planar graphs are exactly the graphs that have neither the complete graph K 5 nor the complete bipartite graph K 3,3 as minors.
Another result relating the four-color theorem to graph minors is the snark theorem announced by Robertson, Sanders, Seymour, and Thomas, a strengthening of the four-color theorem conjectured by W. T. Tutte and stating that any bridgeless 3-regular graph that requires four colors in an edge coloring must have the Petersen graph as a minor. [15]
The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-founded by homeomorphic embeddability, implying such a sequence cannot be infinite. Then, by applying Kőnig's lemma on the tree of such sequences under extension, for each value of k there is a sequence with maximal length.
For example, the Robertson–Seymour theorem guarantees that there is a finite list of forbidden minors that characterizes (for example) the set of graphs that can be embedded on a torus; moreover, Robertson and Seymour showed that there is an O(n 3) algorithm for determining whether a graph has a given graph as a minor.
Paul D. Seymour FRS (born 26 July 1950) is a British mathematician known for his work in discrete mathematics, especially graph theory.He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ ...
In the first of their famous series of papers on graph minors, Neil Robertson and Paul Seymour define a path-decomposition of a graph G to be a sequence of subsets X i of vertices of G, with two properties: For each edge of G, there exists an i such that both endpoints of the edge belong to subset X i, and