Displaying similar documents to “Some totally 4-choosable multigraphs”

An Extension of Kotzig’s Theorem

Valerii A. Aksenov, Oleg V. Borodin, Anna O. Ivanova (2016)

Discussiones Mathematicae Graph Theory

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In 1955, Kotzig proved that every 3-connected planar graph has an edge with the degree sum of its end vertices at most 13, which is tight. An edge uv is of type (i, j) if d(u) ≤ i and d(v) ≤ j. Borodin (1991) proved that every normal plane map contains an edge of one of the types (3, 10), (4, 7), or (5, 6), which is tight. Cole, Kowalik, and Škrekovski (2007) deduced from this result by Borodin that Kotzig’s bound of 13 is valid for all planar graphs with minimum degree δ at least 2...

Acyclic 6-Colouring of Graphs with Maximum Degree 5 and Small Maximum Average Degree

Anna Fiedorowicz (2013)

Discussiones Mathematicae Graph Theory

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A k-colouring of a graph G is a mapping c from the set of vertices of G to the set {1, . . . , k} of colours such that adjacent vertices receive distinct colours. Such a k-colouring is called acyclic, if for every two distinct colours i and j, the subgraph induced by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, every cycle in G has at least three distinct colours. Acyclic colourings were introduced by Gr¨unbaum in 1973, and since...

The structure of plane graphs with independent crossings and its applications to coloring problems

Xin Zhang, Guizhen Liu (2013)

Open Mathematics

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If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity...

Colouring vertices of plane graphs under restrictions given by faces

Július Czap, Stanislav Jendrol' (2009)

Discussiones Mathematicae Graph Theory

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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 colours. We prove that this conjecture is true for 2-connected cubic plane graphs. Next we consider...

On (p, 1)-total labelling of 1-planar graphs

Xin Zhang, Yong Yu, Guizhen Liu (2011)

Open Mathematics

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A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ(G) ≥ 8p+4 or Δ(G) ≥ 6p+2 and g(G) ≥ 4. As a consequence, the well-known (p, 1)-total labelling conjecture has been confirmed for some 1-planar graphs.