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Abstract
A pair (A, B) of square (0, 1)matrices is called a Lehman pair if AB^{T} = 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 nondegenerate 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 6vertex subgraph that we call a 3rung 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 3rung 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.
Original language  English 

Article number  P3.54 
Journal  Electronic Journal of Combinatorics 
Volume  26 
Issue number  3 
Publication status  Published  Sept 2019 
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 1 Finished

Real chromatic roots of graphs and matroids
ARC Australian Research Council
1/01/14 → 31/12/19
Project: Research