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### Graphs with Two Crossings Are 5-Choosable

Zdeněk Dvořák, Bernard Lidický, Riste Škrekovski
2011 SIAM Journal on Discrete Mathematics
We extend the result by showing that every graph with at most two crossings is 5-choosable.  ...  A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. Thomassen's theorem states that every planar graph is 5-choosable.  ...  Since all graphs with crossing number one are 5-choosable by [4] , we can assume that G has two dangerous configurations: either cr(G) = 2 or cr(G) = 1 and G contains T .  ...

### Planar graphs without cycles of length 4, 7, 8, or 9 are 3-choosable

Yingqian Wang, Qian Wu, Liang Shen
2011 Discrete Applied Mathematics
It is known that planar graphs without cycles of length 4, i, j, or 9 with 4 < i < j < 9, except that i = 7 and j = 8, are 3-choosable.  ...  This paper proves that planar graphs without cycles of length 4, 7, 8, or 9 are also 3-choosable.  ...  Acknowledgements The authors are grateful to the referees for their wise suggestions to simplify the original proof of this paper.  ...

### (1,λ)-embedded graphs and the acyclic edge choosability [article]

Xin Zhang, Guizhen Liu, Jian-Liang Wu
2011 arXiv   pre-print
A (1,λ)-embedded graph is a graph that can be embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge.  ...  In this paper, it is shown that every (1,λ)-embedded graph G is 4-linear for all possible λ, and is acyclicly edge-(3Δ(G)+70)-choosable for λ=1,2.  ...  By Theorem 2.2, the following two corollaries are natural. 3 Acyclic edge choosability of (1, λ)-embedded graphs In this section we mainly investigate the acyclic edge choosability of (1, λ)-embedded  ...

### Defective Colouring of Graphs Excluding A Subgraph or Minor

Patrice Ossona De Mendez, Sang-Il Oum, David R. Wood
2018 Combinatorica
This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with relevance to the earth-moon problem), graphs  ...  with given stack- or queue-number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, and graphs excluding a complete bipartite graph as a topological minor.  ...  In particular, Lemma 5.2 implies that graphs with thickness 2 are (5, 36)-choosable, (6, 19) -choosable, (7, 12)-choosable, (8, 9) -choosable, (9, 6)-choosable, (10, 4)-choosable, and (11, 2)-choosable  ...

### Flexibility of planar graphs without 4-cycles [article]

Tomáš Masařík
2019 arXiv   pre-print
Recently, the structural properties of planar graphs in terms of flexibility were investigated. We continue this line of research. Let G be a planar graph with a list assignment L.  ...  The notion called flexibility was recently defined in [Dvořák, Norin, Postle: List coloring with requests, Journal of Graph Theory 2019].  ...  In the latter case two colors are crossed out from the list of v. Keep in mind that this cannot happen twice since there are no two edge-adjacent triangles.  ...

### Choosability in bounded sequential list coloring [article]

Simone Gama, Rosiane de Freitas, Mário Salvatierra
2018 arXiv   pre-print
some classes of graphs, such as complete bipartite graph, which is 3 -choosable, but 2 -(γ,μ)-choosable.  ...  This is a Π_2^P-complete problem, however, we show that k-(γ,μ)-choosability is an NP-problem due to its relation with the k-coloring of a graph and application of methods of proof in choosability for  ...  This method is used to show that every planar graph is 6-choosable and every triangle-free planar graph are 4-choosable [17] .  ...

### Free Choosability of Outerplanar Graphs

Yves Aubry, Jean-Christophe Godin, Olivier Togni
2015 Graphs and Combinatorics
of b colors such that adjacent vertices are colored with disjoint color sets.  ...  A graph G is free (a, b)-choosable if for any vertex v with b colors assigned and for any list of colors of size a associated with each vertex u = v, the coloring can be completed by choosing for u a subset  ...  , DIJON, France Free Choosability of Outerplanar GraphsAn outerplanar graph is a graph that has a crossing-free embedding in the plane such that all vertices are on the same face (without loss of  ...

### (1,λ)-EMBEDDED GRAPHS AND THE ACYCLIC EDGE CHOOSABILITY

Xin Zhang, Guizhen Liu, Jian-Liang Wu
2012 Bulletin of the Korean Mathematical Society
A (1, λ)-embedded graph is a graph that can be embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge.  ...  In this paper, it is shown that every (1, λ)-embedded graph G is 4-linear for all possible λ, and is acyclicly edge-(3∆(G) + 70)-choosable for λ = 1, 2.  ...  investigate the acyclic edge choosability of (1, λ)embedded graphs with special given λ.  ...

### Complexity of choosability with a small palette of colors [article]

Marc Demange, Dominique de Werra
2017 arXiv   pre-print
We also exhibit some classes of graphs (defined by structural properties of their blocks) which are choosable.  ...  We study complexity issues of choosability of graphs when the number k of colors is limited.  ...  edge crossing.  ...

### Defective and clustered choosability of sparse graphs

Kevin Hendrey, David R. Wood
2019 Combinatorics, probability & computing
It implies that every graph with maximum average degree m is $\lfloor{\frac{3}{4}m+1}\rfloor$-choosable with clustering 2.  ...  ., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering.  ...  This research was initiated at the Bellairs Workshop on Graph Theory (20-27 April 2018).  ...

### Critically paintable, choosable or colorable graphs

Ayesha Riasat, Uwe Schauz
2012 Discrete Mathematics
We also show that on a fixed given surface, there are only finitely many critically k-paintable/k-choosable/ k-colorable graphs, if k ≥ 6.  ...  Finally, we use a Ramsey-type lemma to deduce all 2-paintable, 2-choosable, critically 3-paintable and critically 3-choosable graphs, with respect to vertex deletion and to edge deletion.  ...  Graphs on surfaces In this section, we study simple graphs G that are drawn on a surface without crossing edges.  ...

### List edge colourings of some 1-factorable multigraphs

M. N. Ellingham, Luis Goddyn
1996 Combinatorica
whenever e and f are adjacent edges.  ...  The List Edge Colouring Conjecture asserts that, given any multigraph G with chromatic index k and any set system fSe : e 2 E(G)g with each jSej = k, we can choose elements se 2 Se such that se 6 = sf  ...  Two oriented 2-factors i ; j 2 are said to cross at v if the circuits i (v), j (v) geometrically cross at v.  ...

### Path Choosability of Planar Graphs

Glenn G. Chappell, Chris Hartman
2018 Electronic Journal of Combinatorics
For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.  ...  We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths.  ...  Let G be a planar graph.  ...

### Graph Planarity Testing with Hierarchical Embedding Constraints [article]

Giuseppe Liotta, Ignaz Rutter, Alessandra Tappini
2019 arXiv   pre-print
We prove that the problem is fixed-parameter tractable for biconnected graphs, where the parameters are the treewidth of G and the number of FPQ-trees associated with every vertex of G.  ...  Hierarchical embedding constraints define a set of allowed cyclic orders for the edges incident to the vertices of a graph. These constraints are expressed in terms of FPQ-trees.  ...  Figure 5 : 5 Figure 5: (a) A triconnected cubic non-planar graph G with a proper 3-edge-coloring.  ...

### A note on choosability with defect 1 of graphs on surfaces [article]

Vida Dujmović, Djedjiga Outioua
2018 arXiv   pre-print
For example, the chromatic number of the family of toroidal graphs is known to be 7. The bound above implies that toroidal graphs are 5-choosable with defect 1.  ...  This strengthens the result of Cowen, Goddard and Jesurum (1997) who showed that toroidal graphs are 5-colourable with defect 1.  ...  Cowen, Goddard and Jesurum [5, 4] proved that toroidal graphs are (5, 1) * −colourable. eorem 1 implies that they are, in fact, (5, 1) * −choosable thus the theorem constitutes a strengthening of that  ...
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