On interval clutters

A hypergraph is an interval hypergraph if its vertices can be linearly ordered so that all its edges are consecutive sets. Interval hypergraphs have been characterized by Tucker (J. Combin. Theory 12 (1972) 153) in terms of excluded subhypergraphs. In this paper, we strengthen Tucker\u27s result for clutters by characterizing interval clutters in terms of excluded partial clutters, as well as excluded minors. Since minor and partial clutter relations are much more restrictive than the subhypergraph relation, our results are more applicable than Tucker\u27s result in, many situations. As a lemma, we also determine all the minor minimal clutters that have a circuit subhypcrgraph but not a circuit minor. © 2002 Published by Elsevier Science B.V

On Box-Perfect Graphs

Let $G=(V,E)$ be a graph and let $A_G$ be the clique-vertex incidence matrix of $G$. It is well known that $G$ is perfect iff the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call $G$ box-perfect if the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is box-totally dual integral (box-TDI), and posed the problem of characterizing such graphs. In this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity graphs, and identify several other classes of box-perfect graphs. We also develop a general and powerful method for establishing box-perfectness

Excluding a long double path minor

The height of a graph G is defined to be the number of steps to construct G by two simple graph operations. Let Bn be the graph obtained from an n-edge path by doubling each edge in parallel. Then, for any minor-closed class script G of graphs, the following are proved to be equivalent: (1) Some Bn is not in script G; (2) There is a number h such that every graph in script G has height at most h; (3) script G is well-quasi-ordered by the topological minor relation; (4) There is a polynomial function p( • ) such that the number of paths of every graph G in script G is at most p(\V(G)\ + \E(G)\). © 1996 Academic Press, Inc

Clutters with τ2=2τ

Motivated by Lehman\u27s characterization of the minor-minimal clutters without the MFMC property, we propose a conjecture about the minor-minimal clutters with τ

Stable sets versus independent sets

Let G=(V, E) be a graph and let L(G) be the set of stable sets of G. The matroidal number of G, denoted by m(G), is the smallest integer m such that L(G)=J1∪J2∪⋯∪Jm for some matroids Mi=(V,Ji)(i=1,2,...,m). We characterize the graphs of matroidal number at most m for all m≥1. For m≤3, we show that the graphs of matroidal number at most m can be characterized by excluding finitely many induced subgraphs. We also consider a similar problem which replaces \u27union\u27 by \u27intersection\u27. © 1993

Monotone clutters

A clutter is k-monotone, completely monotone or threshold if the corresponding Boolean function is k-monotone, completely monotone or threshold, respectively. A characterization of k-monotone clutters in terms of excluded minors is presented here. This result is used to derive a characterization of 2-monotone matroids and of 3-monotone matroids (which turn out to be all the threshold matroids). © 1993

Disjoint circuits on a Klein bottle and a theorem on posets

In this paper, we consider the problem of packing disjoint directed circuits in a digraph drawn on the Klein bottle or on the torus. We formulate a problem on posets which unifies all the problems considered by Ding et al. and by Seymour. Then we generalize all the results of their two papers by proving a theorem on our special posets. © 1993

The edge version of Hadwiger\u27s conjecture

A well known conjecture of Hadwiger asserts that Kn + 1 is the only minor minimal graph of chromatic number greater than n. In this paper, all minor minimal graphs of chromatic index greater than n are determined. © 2008 Elsevier B.V. All rights reserved