Simultaneously Colouring the Edges and Faces of Plane Graphs

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Simultaneously Colouring the Edges and Faces of Plane Graphs

The elements of a plane graph G are the edges, vertices, and faces of G. We say that two elements are neighbours in G if they are incident with or are mutually adjacent with each other in G. The simultaneous colouring of distinct elements of a planar graph was first introduced by Ringel [12]. In his paper Ringel considered the problem of colouring the vertices and faces of a plane graph in such...

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Colouring vertices of plane graphs under restrictions given by faces

We consider a vertex colouring of a connected plane graph G. A colour c is used k times by a face α of G if it appears k times along the facial walk of α. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such a colouring can be done using three c...

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Colouring of plane graphs with unique maximal colours on faces

The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colours. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1, . . . , 4 in such a way that every face contains a unique vertex coloured with the maximal colour appearing on that face. The...

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Edge-colouring planar graphs with precoloured edges

Let G be a simple planar graph of maximum degree ∆, and let H be a nonempty subgraph of G that has been (∆+t)-edge-coloured. We prove that if H has maximum degree d ≤ t, and ∆ is large enough, the edge-precolouring can be extended to a (∆ + t)-edge-colouring of G. If d > t, there are examples for any choice of ∆ where the extension is impossible. When d = t, large enough means ∆ ≥ 16 + d, when ...

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Colouring proximity graphs in the plane

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ژورنال

عنوان ژورنال: Journal of Combinatorial Theory, Series B

سال: 1997

ISSN: 0095-8956

DOI: 10.1006/jctb.1997.1725