Structure of Cubic Lehman Matrices

A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called \emph{Lehman matrices}. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs} (also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, L\"{u}tolf \& Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

An extension of Lehman's theorem and ideal set functions

Lehman’s theorem on the structure of minimally nonideal clutters is a fundamental result in polyhedral combinatorics. One approach to extending it has been to give a common generalization with the characterization of minimally imperfect clutters (Sebő, 1998; Gasparyan et al., 2003). We give a new generalization of this kind, which combines two types of covering inequalities and works well with the natural definition of minors. We also show how to extend the notion of idealness to unit-increasing set functions, in a way that is compatible with minors and blocking operations

LEHMAN MATRICES

A pair of square 0, 1 matrices A, B such that ABT = E + kI (where E is the n × n matrix of all 1s and k is a positive integer) are called Lehman matrices. These matrices figure prominently in Lehman’s seminal theorem on minimally nonideal matrices. There are two choices of k for which this matrix equation is known to have infinite families of solutions. When n = k2 +k +1 and A = B, we get point-line incidence matrices of finite projective planes, which have been widely studied in the literature. The other case occurs when k = 1 and n is arbitrary, but very little is known in this case. This paper studies this class of Lehman matrices and classifies them according to their similarity to circulant matrices

Structure of cubic lehman matrices

A pair (A, B) of square (0, 1)-matrices is called a Lehman pair if ABT = J + kI for some integer k ∈ {−1, 1, 2, 3, …}. In this case A and B are called Lehman matrices. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix M form a square Lehman submatrix of M. Lehman matrices with k = −1 are essentially equivalent to partitionable graphs (also known as (α, ω)-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular 6-vertex subgraph that we call a 3-rung ladder segment. Two decades ago, Lütolf & Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with k = 1 of order up to 17 × 17. We verify their catalogue (which has just one omission), and extend the computational results to 20 × 20 matrices. Of the 908 cubic Lehman matrices (with k = 1) of order up to 20 × 20, only two do not arise from our 3-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with k = 1