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Graphs with Two Crossings Are 5-Choosable

2011
*
SIAM Journal on Discrete Mathematics
*

We extend the result by showing that every

doi:10.1137/11082703x
fatcat:ogxvkmqtmndh3gdiuavcmjbdxa
*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 . ...##
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Planar graphs without cycles of length 4, 7, 8, or 9 are 3-choosable

2011
*
Discrete Applied Mathematics
*

It is known that planar

doi:10.1016/j.dam.2010.11.002
fatcat:g6hk5nmh6rhjnaefrrxjcnwmlm
*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. ...##
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(1,λ)-embedded graphs and the acyclic edge choosability
[article]

2011
*
arXiv
*
pre-print

A (1,λ)-embedded

arXiv:1106.4681v2
fatcat:jlj743cpqjesdpcsb5mv2rifza
*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 ...##
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Defective Colouring of Graphs Excluding A Subgraph or Minor

2018
*
Combinatorica
*

This result leads to new defective colouring results for several

doi:10.1007/s00493-018-3733-1
fatcat:b3gr6so5vragxdjmlbap3rw3mm
*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*...##
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Flexibility of planar graphs without 4-cycles
[article]

2019
*
arXiv
*
pre-print

Recently, the structural properties of planar

arXiv:1903.01460v1
fatcat:7smpun3x75evxk6dhehlrwffly
*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. ...##
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Choosability in bounded sequential list coloring
[article]

2018
*
arXiv
*
pre-print

some classes of

arXiv:1812.11685v1
fatcat:c2bryajixjdjrhhgfxsfugtg4i
*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] . ...##
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Free Choosability of Outerplanar Graphs

2015
*
Graphs and Combinatorics
*

of b colors such that adjacent vertices

doi:10.1007/s00373-015-1625-3
fatcat:awu4ljmev5d3rhs7cqozsd6mcm
*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 ...##
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(1,λ)-EMBEDDED GRAPHS AND THE ACYCLIC EDGE CHOOSABILITY

2012
*
Bulletin of the Korean Mathematical Society
*

A (1, λ)-embedded

doi:10.4134/bkms.2012.49.3.573
fatcat:pobqygwksnacjmwskr5wu7mq5i
*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. ...##
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Complexity of choosability with a small palette of colors
[article]

2017
*
arXiv
*
pre-print

We also exhibit some classes of

arXiv:1601.01768v2
fatcat:xk57pcfvc5edho7good4b6uytm
*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*. ...##
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Defective and clustered choosability of sparse graphs

2019
*
Combinatorics, probability & computing
*

It implies that every

doi:10.1017/s0963548319000063
fatcat:z4w4ci523rfe5ketzigqyzwphi
*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). ...##
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Critically paintable, choosable or colorable graphs

2012
*
Discrete Mathematics
*

We also show that on a fixed given surface, there

doi:10.1016/j.disc.2012.07.035
fatcat:esfrt5d2qvarja6h7wobmklnai
*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. ...##
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List edge colourings of some 1-factorable multigraphs

1996
*
Combinatorica
*

whenever e and f

doi:10.1007/bf01261320
fatcat:hv4x3jwatjdrzfduwbespn6dia
*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. ...##
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Path Choosability of Planar Graphs

2018
*
Electronic Journal of Combinatorics
*

For planar

doi:10.37236/7139
fatcat:gt7nz5bslffdpnrcbtbc7ta65u
*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*. ...##
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Graph Planarity Testing with Hierarchical Embedding Constraints
[article]

2019
*
arXiv
*
pre-print

We prove that the problem is fixed-parameter tractable for biconnected

arXiv:1904.12596v3
fatcat:takkfqjfj5bbharavej4qr62wa
*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. ...##
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A note on choosability with defect 1 of graphs on surfaces
[article]

2018
*
arXiv
*
pre-print

For example, the chromatic number of the family of toroidal

arXiv:1806.06149v1
fatcat:edcaabggpzcsjkqwojnyinemd4
*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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