Alternating Signed Bipartite Graphs and Difference-1 Colourings

TL;DR

This paper explores the conditions under which bipartite graphs can be coloured to form alternating signed bipartite graphs, introducing difference-1 colourings as a relaxed concept and generalising Hall's Matching Theorem.

Contribution

It introduces the concept of difference-1 colourings, provides necessary and sufficient conditions for their existence, and generalises Hall's Matching Theorem for bipartite graphs.

Findings

Characterisation of when a bipartite graph admits a difference-1 colouring

Identification of graph classes where all difference-1 colourings are ASBG-colourings

A generalisation of Hall's Matching Theorem for prescribed degree subgraphs

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 $r(v)$.