{"work":{"id":"851d137a-d6cb-4da0-8935-9a63ddbf861d","openalex_id":"https://openalex.org/W1977465442","doi":"10.1176/appi.ajp.2010.09091379","arxiv_id":"2010.09091","raw_key":null,"title":"Colourings of $(m, n)$-coloured mixed graphs","authors":null,"authors_text":"Thomas R. Insel, Bruce N. Cuthbert, Marjorie A. Garvey, Robert Heinssen, Daniel S. Pine, Kevin J. Quinn, Charles A. Sanislow, Philip Wang","year":2020,"venue":"math.CO","abstract":"A mixed graph is, informally, an object obtained from a simple undirected graph by choosing an orientation for a subset of its edges. A mixed graph is $(m, n)$-coloured if each edge is assigned one of $m \\geq 0$ colours, and each arc is assigned one of $n \\geq 0$ colours. Oriented graphs are $(0, 1)$-coloured mixed graphs, and 2-edge-coloured graphs are $(2, 0)$-coloured mixed graphs. We show that results of Sopena for vertex colourings of oriented graphs, and of Kostochka, Sopena and Zhu for vertex colourings oriented graphs and 2-edge-coloured graphs, are special cases of results about vertex colourings of $(m, n)$-coloured mixed graphs. Both of these can be regarded as a version of Brooks' Theorem.","external_url":"https://arxiv.org/abs/2010.09091","cited_by_count":7056,"metadata_source":"pith","metadata_fetched_at":"2026-08-05T02:28:24.338817+00:00","pith_arxiv_id":"2010.09091","created_at":"2026-07-07T20:04:04.163268+00:00","updated_at":"2026-08-05T02:49:54.815029+00:00","title_quality_ok":true,"display_title":null,"render_title":"Colourings of $(m, n)$-coloured mixed graphs"},"hub":{"state":{"tier_text":"hub","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":1,"external_cited_by_count":7056},"tier":"hub","role_counts":[],"polarity_counts":[],"runs":{},"summary":{},"graph":{},"authors":[]}}