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Discrete mathematics and mathematical cybernetics
Vertex colourings of multigraphs with forbiddances on edges
A. N. Glebova, I. A. Pavlovb, K. A. Khadaevc a Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
b Novosibirsk State University, 2, Pirogova str., Novosibirsk, 630090, Russia
c Higher School of Economics, 20, Myasnitskaya str., Moscow, 101000, Russia
Abstract:
We define and study a new class of vertex colourings of multigraphs, where some pairs of colours are forbidden on the edges of a multigraph. We say that a multigraph $G$ is (properly) $(m,r)$-colourable if for any given sets of $r$ forbidden pairs of colours on the edges of $G$ where exists a (proper) vertex $m$-colouring of $G$ that respects all forbidden pairs. We determine all (properly) $(m,r)$-colourable stars, all $(2,r)$-colourable multigraphs for each $r\ge 1$ and all $(m,r)$-colourable multighraphs, where $r$ is large enough (close to $m^2$). We also introduce a list version of $(m,r)$-colourability and establish (for the case of improper colourings) that the list $(m,r)$-colourability of a multigraph is equivalent to its $(m,r)$-colourability.
Keywords:
graph, multigraph, edge, colouring, list colouring, forbiddance.
Received November 3, 2018, published April 24, 2020
Citation:
A. N. Glebov, I. A. Pavlov, K. A. Khadaev, “Vertex colourings of multigraphs with forbiddances on edges”, Sib. Èlektron. Mat. Izv., 17 (2020), 637–646
Linking options:
https://www.mathnet.ru/eng/semr1237 https://www.mathnet.ru/eng/semr/v17/p637
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Abstract page: | 165 | Full-text PDF : | 100 | References: | 11 |
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