Abstract

We investigate a class of 2-edge coloured bipartite graphs known as alternating signed bipartite graphs (ASBGs) that encode the information in alternating sign matrices. The central question is when a given bipartite graph admits an ASBG-colouring; a 2-edge colouring such that the resulting graph is an ASBG. We introduce the concept of a difference-1 colouring, a relaxation of the concept of an ASBG-colouring, and present a set of necessary and sufficient conditions for when a graph admits a difference-1 colouring. The relationship between distinct difference-1 colourings of a particular graph is characterised, and some classes of graphs for which all difference-1 colourings are ASBG-colourings are identified. One key step is Theorem 3.4.6, which generalises Hall's Matching Theorem by describing a necessary and sufficient condition for the existence of a subgraph H of a bipartite graph in which each vertex v of H has some prescribed degree r(v).

Highlights

  • We develop a theme introduced by Brualdi, Kiernan, Meyer, and Schroeder in [1], by investigating a class of bipartite graphs related to alternating sign matrices

  • Recent developments in the study of alternating sign matrix (ASM) include an investigation of their spectral properties in [9], an extension of the concept of Latin square arising from the replacement of permutation matrices by ASMs in [10], and a study of some graphs arising from ASMs in [1]

  • This latter article introduces the concept of the alternating signed bipartite graph (ASBG) of an ASM, which is constructed from an alternating sign matrix as follows

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Summary

Introduction

We develop a theme introduced by Brualdi, Kiernan, Meyer, and Schroeder in [1], by investigating a class of bipartite graphs related to alternating sign matrices. We may associate to any matrix the bipartite graph whose vertices correspond to rows and columns, and where an edge between the vertices representing Row i and Column j encodes the information that the (i, j) entry is non-zero. Additional features of the non-zero entries (for example, sign) might be indicated by assigning colours to the edges. In the case of alternating sign matrices, the special matrix structure translates to particular combinatorial properties of the resulting bipartite graphs. We introduce the main objects of interest in this opening section

Alternating sign matrices and alternating signed bipartite graphs
Obstacles to ASBG-colourability
Difference-1 colourings
Configurability for cactus graphs
Deciding the existence of a difference-1 colouring
Unicyclic graphs
Graphs with junctions
Redistributability
Uniqueness and difference-k colourings
Uniqueness of difference-1 colourings
Difference-k colourings
Full Text
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