Abstract not found

A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first

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.

We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture, and Pfaffian orientations of bipartite graphs.

A graph is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either |C|≥4, or G has an edge with one end in A and the other end in B, orone of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P − 10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte’s Four Flow Conjecture: every 2-edge connected graph with no minor isomorphic to P − 10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.