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The Robertson–Seymour theorem states that finite undirected graphs and graph minors form a well-quasi-ordering. The graph minor relationship does not contain any infinite descending chain, because each contraction or deletion reduces the number of edges and vertices of the graph (a non-negative integer). [8]
A graph has tree width at most k if it can be obtained via k-clique-sums from a list of graphs, where each graph in the list has at most k + 1 vertices. Corollary 1 indicates to us that k -clique-sums of small graphs describe the rough structure H -free graphs when H is planar.
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.
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.
Pages in category "Graph minor theory" The following 33 pages are in this category, out of 33 total. ... Robertson–Seymour theorem; S. Shallow minor; Snark (graph ...
The Robertson–Seymour theorem implies that every matroid property of graphic matroids characterized by a list of forbidden minors can be characterized by a finite list. Another way of saying the same thing is that the partial order on graphic matroids formed by the minor operation is a well-quasi-ordering. However, the example of the real ...
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In 1993, with Seymour and Robin Thomas, Robertson proved the -free case for which the Hadwiger conjecture relating graph coloring to graph minors is known to be true. [ 8 ] In 1996, Robertson, Seymour, Thomas, and Daniel P. Sanders published a new proof of the four color theorem , [ 9 ] confirming the Appel–Haken proof which until then had ...