Odd cycles and matrices with integrality properties

Abstract

A cycle of a bipartite graphG(V⁺, V⁻; E) is odd if its length is 2 (mod 4), even otherwise. An odd cycleC is node minimal if there is no odd cycleC′ of cardinality less than that ofC′ such that one of the following holds:C′ ∩V ⁺ ⊂C ∩V ⁺ orC′ ∩V ⁻ ⊂C ∩V ⁻.

In this paper we prove the following theorem for bipartite graphs:

For a bipartite graphG, one of the following alternatives holds:

• -All the cycles ofG are even.

• -G has an odd chordless cycle.

• -For every node minimal odd cycleC, there exist four nodes inC inducing a cycle of length four.

• -An edge (u, v) ofG has the property that the removal ofu, v and their adjacent nodes disconnects the graphG.

To every (0, 1) matrixA we can associate a bipartite graphG(V⁺, V⁻; E), whereV ⁺ andV ⁻ represent respectively the row set and the column set ofA and an edge (i,j) belongs toE if and only ifa _ij = 1. The above theorem, applied to the graphG(V⁺, V⁻; E) can be used to show several properties of some classes of balanced and perfect matrices. In particular it implies a decomposition theorem for balanced matrices containing a node minimal odd cycleC, having the property that no four nodes ofC induce a cycle of length 4. The above theorem also yields a proof of the validity of the Strong Perfect Graph Conjecture for graphs that do not containK ₄−e as an induced subgraph.