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Some examples of finite obstruction sets were already known for specific classes of graphs before the Robertson–Seymour theorem was proved. For example, the obstruction for the set of all forests is the loop graph (or, if one restricts to simple graphs, the cycle with three vertices). This means that a graph is a forest if and only if none of ...
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.
Robertson's example of a non-YΔY-reducible apex graph. A connected graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of steps, each of which is a Δ-Y or Y-Δ transform , the removal of a self-loop or multiple adjacency, the removal of a vertex with one neighbor, and the replacement of a vertex of degree two and its ...
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]
Robertson–Seymour theorem; Petersen graph; Planar graph. Dual polyhedron; Outerplanar graph; Random graph; Regular graph; Scale-free network; Snark (graph theory) Sparse graph. Sparse graph code; Split graph; String graph; Strongly regular graph; Threshold graph; Total graph; Tree (graph theory). Trellis (graph) Turán graph; Ultrahomogeneous ...
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]
A version of this theorem proved by Wagner (1937) states that if a graph G is both K 5-free and K 3,3-free, then G is planar. This theorem provides a "good reason" for a graph G not to have K 5 or K 3,3 as minors; specifically, G embeds on the sphere, whereas neither K 5 nor K 3,3 embed on the sphere.
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 ...