Combinatorics in Oxford in the 1970s was dominated by matroid theory, due to the influence of Dominic Welsh and Aubrey William Ingleton. Much of Seymour's early work, up to about 1980, was on matroid theory, and included three important matroid results: his D.Phil. thesis on matroids with the max-flow min-cut property[pub 1] (for which he won his first Fulkerson prize); a characterisation by excluded minors of the matroids representable over the three-element field;[pub 2] and a theorem that all regular matroids consist of graphic and cographic matroids pieced together in a simple way[pub 3] (which won his first Pólya prize). There were several other significant papers from this period: a paper with Welsh on the critical probabilities for bond percolation on the square lattice;[pub 4] a paper on edge-multicolouring of cubic graphs,[pub 5] which foreshadows the matching lattice theorem of László Lovász; a paper proving that all bridgeless graphs admit nowhere-zero 6-flows,[pub 6] a step towards Tutte's nowhere-zero 5-flow conjecture; and a paper solving the two-paths problem (also introducing the cycle double cover conjecture),[pub 7] which was the engine behind much of Seymour's future work.
In 1980 he moved to Ohio State University, and began work with Neil Robertson. This led eventually to Seymour's most important accomplishment, the so-called "Graph Minors Project", a series of 23 papers (joint with Robertson), published over the next thirty years, with several significant results:
the graph minors structure theorem, that for any fixed graph, all graphs that do not contain it as a minor can be built from graphs that are essentially of bounded genus by piecing them together at small cutsets in a tree structure;[pub 8]
a proof of a conjecture of Wagner that in any infinite set of graphs, one of them is a minor of another (and consequently that any property of graphs that can be characterised by excluded minors can be characterised by a finite list of excluded minors);[pub 9]
a proof of a similar conjecture of Nash-Williams that in any infinite set of graphs, one of them can be immersed in another;[pub 10]
and polynomial-time algorithms to test if a graph contains a fixed graph as a minor, and to solve the k vertex-disjoint paths problem for all fixed k.[pub 11]
In about 1990 Robin Thomas began to work with Robertson and Seymour. Their collaboration resulted in several important joint papers over the next ten years:
a proof of a conjecture of Sachs, characterising by excluded minors the graphs that admit linkless embeddings in 3-space;[pub 12]
a proof that every graph that is not five-colourable has a six-vertex complete graph as a minor (the four-colour theorem is assumed to obtain this result, which is a case of Hadwiger's conjecture);[pub 13]
with Dan Sanders, a new, simplified, computer based proof of the four-colour theorem;[pub 14]
and a description of the bipartite graphs that admit Pfaffian orientations.[pub 15]
In the same period, Seymour and Thomas also published several significant results: (with Noga Alon) a separator theorem for graphs with an excluded minor,[pub 16] extending the planar separator theorem of Richard Lipton and Robert Tarjan; a paper characterizing treewidth in terms of brambles;[pub 17] and a polynomial-time algorithm to compute the branch-width of planar graphs.[pub 18]
In 2000 Robertson, Seymour, and Thomas were supported by the American Institute of Mathematics to work on the strong perfect graph conjecture, a famous open question that had been raised by Claude Berge in the early 1960s. Seymour's student Maria Chudnovsky joined them in 2001, and in 2002 the four jointly proved the conjecture.[pub 19] Seymour continued to work with Chudnovsky, and obtained several more results about induced subgraphs, in particular (with Cornuéjols, Liu, and Vušković) a polynomial-time algorithm to test whether a graph is perfect,[pub 20] and a general description of all claw-free graphs.[pub 21] Other important results in this period include: (with Seymour's student Sang-il Oum) fixed-parameter tractable algorithms to approximate the clique-width of graphs (within an exponential bound) and the branch-width of matroids (within a linear bound);[pub 22] and (with Chudnovsky) a proof that the roots of the independence polynomial of every claw-free graph are real.[pub 23]
In the 2010s Seymour worked mainly on χ-boundedness and the Erdős–Hajnal conjecture. In a series of papers with Alex Scott and partly with Chudnovsky, they proved two conjectures of András Gyárfás, that every graph with bounded clique number and sufficiently large chromatic number has an induced cycle of odd length at least five,[pub 24] and has an induced cycle of length at least any specified number.[pub 25] The series culminated in a paper of Scott and Seymour proving that for every fixed k, every graph with sufficiently large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k,[pub 26] which leads to the resolutions of two conjectures of Gil Kalai and Roy Meshulam connecting the chromatic number of a graph with the homology of its independence complex. There was also a polynomial-time algorithm (with Chudnovsky, Scott, and Chudnovsky and Seymour's student Sophie Spirkl) to test whether a graph contains an induced cycle with length more than three and odd.[pub 27] Most recently, the four jointly resolved the 5-cycle case of the Erdős–Hajnal conjecture, which says that every graph without an induced copy of the 5-cycle contains an independent set or a clique of polynomial size.[pub 28]
