84 research outputs found
Circular choosability
International audienceWe study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O(ch(G) + ln |V(G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f-choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of τ := sup {cch(G) : G is planar}, and we prove that 68, thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density
Circular Choosability
In this paper, we study the notion of circular choosability recently introduced by Mohar and Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that, for every graph G, cch(G) = O( ch(G) + ln |V(G)| ). We investigate a generalisation of circular choosability, circular f-choosability, when f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of tau := sup{ cch(G) : G is planar }, and we prove that 6 <= tau <= 8, thereby providing a negative answer to another question of Mohar. Finally, we study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density
Circular choosability is rational
AbstractThe circular choosability or circular list chromatic number of a graph is a list-version of the circular chromatic number, that was introduced by Mohar in 2002 and has been studied by several groups of authors since then. One of the nice properties that the circular chromatic number enjoys is that it is a rational number for all finite graphs G, and a fundamental question, posed by Zhu and reiterated by others, is whether the same holds for the circular choosability. In this paper we show that this is indeed the case
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Few Long Lists for Edge Choosability of Planar Cubic Graphs
It is known that every loopless cubic graph is 4-edge choosable. We prove the
following strengthened result.
Let G be a planar cubic graph having b cut-edges. There exists a set F of at
most 5b/2 edges of G with the following property. For any function L which
assigns to each edge of F a set of 4 colours and which assigns to each edge in
E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the
colour of each edge e belongs to L(e).Comment: 14 pages, 1 figur
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