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Choosability of planar graphs

Margit Voigt
1996 Discrete Mathematics  
We prove the equivalence of the well-known conjecture of Erd6s et al. (1979): "Every planar graph is 5-choosable" with the following conjecture: "Every planar graph is free 5-choosable".  ...  We say G is k-choosable if there is at least one L-list colouring for every possible list assignment L with IL(v)l = k Vv E V(G).  ...  The following conjectures are equivalent: (1) every planar 9raph is 5-choosable, (2) every planar graph is free 5-choosable.  ... 
doi:10.1016/0012-365x(95)00216-j fatcat:gbmir3e765dudcxmed2vanh5fu

A not 3-choosable planar graph without 3-cycles

Margit Voigt
1995 Discrete Mathematics  
Recently, Thomassen has proved that every planar graph with girth greater than 4 is 3-choosable. Furthermore, it is known that the chromatic number of a planar graph without 3-cycles is at most 3.  ...  Consequently, the question resulted whether every planar graph without 3-cycles is 3-choosable. In the following we will give a planar graph without 3-cycles which is not 3choosable.  ...  [2] , namely 'Every planar graph is 5-choosable' 0012-365X/95/$09.50 © 1995--Elsevier Science B.V.  ... 
doi:10.1016/0012-365x(94)00180-9 fatcat:4za5p6ecdngmfkrrsdgolub37e

Colouring planar graphs with bounded monochromatic components

A. N. Glebov
2020 Sibirskie Elektronnye Matematicheskie Izvestiya  
Our second result states that every planar graph of girth 5 is 2-choosable so that each monochromatic component is a tree with at most 515 vertices.  ...  Axenovich et al. proved that every planar graph of girth 6 is 2-choosable so that each monochromatic component is a path with at most 15 vertices.  ...  Theorem 3 . 3 Every graph G with Arb f (G) ≤ 5/3 (and hence every planar graph of girth at least 5) is acyclically 2-choosable with clustering 515.  ... 
doi:10.33048/semi.2020.17.032 fatcat:6ttcqtztnjacfcqgmfzwr6k4uy

A non-3-choosable planar graph without cycles of length 4 and 5

M. Voigt
2007 Discrete Mathematics  
Steinberg's question from 1975 whether every planar graph without 4-and 5-cycles is 3-colorable is still open.  ...  In this paper the analogous question for 3-choosability of such graphs is answered to the negative.  ...  In 1995 it was proved that every planar graph of girth 5 is 3-choosable [10] but this is not true for the class of all planar graphs of girth 4 [11] .  ... 
doi:10.1016/j.disc.2005.11.041 fatcat:r6xnyme46bh4fik7smml6uvrzy

List colourings of planar graphs

Margit Voigt
2006 Discrete Mathematics  
There are two classical conjectures from Erdős, Rubin and Taylor 1979 about the choosability of planar graphs: (1) every planar graph is 5-choosable and, (2) there are planar graphs which are not 4-choosable  ...  A graph G = G(V , E) is called L-list colourable if there is a vertex colouring of G in which the colour assigned to a vertex v is chosen from a list L(v) associated with this vertex.  ...  However, the other conjecture dating from 1979 that every planar graph is 5-choosable remains an open problem.  ... 
doi:10.1016/j.disc.2006.03.027 fatcat:ciclxvodwbgqzf37lnjnan5lle

Acyclic improper choosability of graphs

Louis Esperet, Alexandre Pinlou
2007 Electronic Notes in Discrete Mathematics  
We finally prove that acyclic choosability and acyclic improper choosability of planar graphs are equivalent notions.  ...  Using a linear time algorithm, we also prove that outerplanar graphs are acyclically (2, 5) * -choosable (i.e. they are acyclically 2-choosable with color classes of maximum degree five).  ...  [4] proved that every planar graph is acyclically 7-choosable. They also conjectured that every planar graph is acyclically 5-choosable.  ... 
doi:10.1016/j.endm.2007.01.037 fatcat:hzwkfk5my5ecvk7oojet4h37eu

A refinement of choosability of graphs [article]

Xuding Zhu
2019 arXiv   pre-print
4-colourable implies that every planar graph is {1,1,2}-choosable.  ...  In particular, it is proved that a conjecture of Kündgen and Ramamurthi on list colouring of planar graphs is implied by the conjecture that every planar graph is {2,2}-choosable, and also implied by the  ...  Conjecture 5 [17] Every planar graph is weakly 4-choosable. It is easy to see that Conjecture 1 implies Conjecture 5.  ... 
arXiv:1811.08587v2 fatcat:24rnovkq7ndbjjd2n6lpkzobee

A note on the not 3-choosability of some families of planar graphs

Mickaël Montassier
2006 Information Processing Letters  
Voigt, A non-3choosable planar graph without cycles of length 4 and 5, 2003, Manuscript], Voigt gave a planar graph without 3-cycles and a planar graph without 4-cycles and 5-cycles which are not 3-choosable  ...  A graph G is L-list colorable if for a given list assignment In [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325-328] and [M.  ...  Every planar graph without cycles of length 4, 5, 6, is 3-choosable. We conjecture:  ... 
doi:10.1016/j.ipl.2005.10.014 fatcat:3pxicggk4ffbhluidpyeaqw2ru

Multiple list colouring triangle-free planar graphs [article]

Yiting Jiang, Xuding Zhu
2018 arXiv   pre-print
This paper proves that for each positive integer $m$, there is a triangle-free planar graph $G$ which is not $(3m+ \lceil \frac m{17} \rceil-1, m)$-choosable.  ...  Question 5 Is it true that every triangle free planar graph is (7, 2)-choosable? It is known [10] that for any finite graph G, ch * f (G) is a rational number.  ...  A natural question is whether there is a constant m such that every triangle free planar graph G is (3m, m)-choosable.  ... 
arXiv:1809.03665v2 fatcat:modkewvqm5gknjrxujn5s2we4u

The complexity of planar graph choosability [article]

Shai Gutner
2008 arXiv   pre-print
We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.  ...  In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable.  ...  GRAPH (2,3)-CHOOSABILITY (BG (2,3)-CH) INSTANCE: A bipartite graph G = (V, E) and a function f : V → {2, 3}. Every planar graph is 5-choosable.  ... 
arXiv:0802.2668v1 fatcat:4lkaviaj6rh3zbins3vj32wlbi

Multiple list colouring of planar graphs [article]

Xuding Zhu
2016 arXiv   pre-print
This paper proves that for each positive integer m, there is a planar graph G which is not (4m+2m-1/9,m)-choosable. Then we pose some conjectures concerning multiple list colouring of planar graphs.  ...  Some open problems Thomassen proved that every planar graph is 5-choosable [7] . The proof can be easily adopted to show that for any positive integer m, every planar graph is (5m, m)-choosable.  ...  Without using the Four Colour Theorem, it is proved very recently by Cranston and Rabern [2] that every planar graph is (9, 2)-colourable (an earlier result in [5] shows that every planar graph G is  ... 
arXiv:1605.04690v1 fatcat:azjwxgipobhtnnj4dnukxeecqu

List colourings of planar graphs

Margit Voigt
1993 Discrete Mathematics  
There are two classical conjectures from Erd&, Rubin and Taylor 1979 about the choosability of planar graphs: (1) every planar graph is 5-choosable and, (2) there are planar graphs which are not 4-choosable  ...  A graph G = G( V, E) is called L-list colourable if there is a vertex colouring of G in which the colour assigned to a vertex u is chosen from a list L(v) associated with this vertex.  ...  It is easy to see that every planar graph is 6-choosable and Alon and Tarsi [2] showed that every planar bipartite graph is 3-choosable. (1) Every planar graph is 5-choosable. (2) There are planar graphs  ... 
doi:10.1016/0012-365x(93)90579-i fatcat:2t25kqgjerbipeaayc7g5nhbva

The complexity of planar graph choosability

Shai Gutner
1996 Discrete Mathematics  
We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.  ...  In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable.  ...  Every planar 9raph is 5-choosable. Theorem !.7. There exists a planar 9raph with 75 vertices which is not 4-choosable. Theorem 1.8.  ... 
doi:10.1016/0012-365x(95)00104-5 fatcat:4btk53rlufaunc5x7hsugdonoi

Total choosability of planar graphs with maximum degree 4

Nicolas Roussel
2011 Discrete Applied Mathematics  
Let G be a planar graph with maximum degree 4. It is known that G is 8-totally choosable. It has been recently proved that if G has girth g ⩾ 6, then G is 5-totally choosable.  ...  In this note we improve the first result by showing that G is 7-totally choosable and complete the latter one by showing that G is 6-totally choosable if G has girth at least 5.  ...  Borodin et al. proved [2] that every planar graph with maximum degree 4 is 8-totally choosable.  ... 
doi:10.1016/j.dam.2010.10.001 fatcat:k3nxloju6vdqlos3hosfu7tjhm

Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable—A short proof

Nathann Cohen, Frédéric Havet
2010 Discrete Mathematics  
Every planar graph with maximum degree ∆ ≥ 9 is (∆ + 1)-edge-choosable.  ...  Every planar graph with maximum degree ∆ ≥ 9 is (∆ + 1)-edge-choosable. This work was partially supported by the INRIA associated team EWIN between Mascotte and ParGO.  ...  Conjecture 3 is still open for planar graphs of maximum degree between 5 and 8 and it is still unknown if planar graphs of maximum degree ∆ are ∆-edge-choosable for 6 ≤ ∆ ≤ 11.  ... 
doi:10.1016/j.disc.2010.07.004 fatcat:obvauryyovah7i6ovjpsf33pz4
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