Recent Excluded Minor Theorems for Graphs

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3-space, Hadwiger’s conjecture on t-colorability of graphs with no Kt+1 minor, Tutte’s edge 3-coloring conjecture on edge 3-colorability of 2-connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2-colorability of hypergraphs, and sign-nonsingular matrices.