2025-09-032025-09-032020-07-13O'Brien, C., Jennings, K. and Quinlan, R. (2020) 'Alternating signed bipartite graphs and difference-1 colourings', Linear Algebra and its Applications, 604, 370-398, available: https://doi.org/10.1016/j.laa.2020.06.030.0024-3795https://dspace.mic.ul.ie/handle/10395/3469We 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).engOpen Access CC BY 4.0 Attribution 4.0 International Deedhttps://www.sciencedirect.com/science/article/pii/S0024379520303219?via%3DihubAlternating sign matrixASMBipartite graphEdge weightEdge coloringArticle10.1016/j.laa.2020.06.030