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Choosability of planar graphs

1996
*
Discrete Mathematics
*

We prove the equivalence of the well-known conjecture of Erd6s et al. (1979): "

doi:10.1016/0012-365x(95)00216-j
fatcat:gbmir3e765dudcxmed2vanh5fu
*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*. ...##
###
A not 3-choosable planar graph without 3-cycles

1995
*
Discrete Mathematics
*

Recently, Thomassen has proved that

doi:10.1016/0012-365x(94)00180-9
fatcat:4za5p6ecdngmfkrrsdgolub37e
*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. ...##
###
Colouring planar graphs with bounded monochromatic components

2020
*
Sibirskie Elektronnye Matematicheskie Izvestiya
*

Our second result states that

doi:10.33048/semi.2020.17.032
fatcat:6ttcqtztnjacfcqgmfzwr6k4uy
*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. ...##
###
A non-3-choosable planar graph without cycles of length 4 and 5

2007
*
Discrete Mathematics
*

Steinberg's question from 1975 whether

doi:10.1016/j.disc.2005.11.041
fatcat:r6xnyme46bh4fik7smml6uvrzy
*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] . ...##
###
List colourings of planar graphs

2006
*
Discrete Mathematics
*

There are two classical conjectures from Erdős, Rubin and Taylor 1979 about the

doi:10.1016/j.disc.2006.03.027
fatcat:ciclxvodwbgqzf37lnjnan5lle
*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. ...##
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Acyclic improper choosability of graphs

2007
*
Electronic Notes in Discrete Mathematics
*

We finally prove that acyclic

doi:10.1016/j.endm.2007.01.037
fatcat:hzwkfk5my5ecvk7oojet4h37eu
*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*. ...##
###
A refinement of choosability of graphs
[article]

2019
*
arXiv
*
pre-print

4-colourable implies that

arXiv:1811.08587v2
fatcat:24rnovkq7ndbjjd2n6lpkzobee
*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*. ...##
###
A note on the not 3-choosability of some families of planar graphs

2006
*
Information Processing Letters
*

Voigt, A non-3choosable

doi:10.1016/j.ipl.2005.10.014
fatcat:3pxicggk4ffbhluidpyeaqw2ru
*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: ...##
###
Multiple list colouring triangle-free planar graphs
[article]

2018
*
arXiv
*
pre-print

This paper proves that for each positive integer $m$, there

arXiv:1809.03665v2
fatcat:modkewvqm5gknjrxujn5s2we4u
*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*. ...##
###
The complexity of planar graph choosability
[article]

2008
*
arXiv
*
pre-print

We also obtain simple constructions of a

arXiv:0802.2668v1
fatcat:4lkaviaj6rh3zbins3vj32wlbi
*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*. ...##
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Multiple list colouring of planar graphs
[article]

2016
*
arXiv
*
pre-print

This paper proves that for each positive integer m, there

arXiv:1605.04690v1
fatcat:azjwxgipobhtnnj4dnukxeecqu
*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*...##
###
List colourings of planar graphs

1993
*
Discrete Mathematics
*

There are two classical conjectures from Erd&, Rubin and Taylor 1979 about the

doi:10.1016/0012-365x(93)90579-i
fatcat:2t25kqgjerbipeaayc7g5nhbva
*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*...##
###
The complexity of planar graph choosability

1996
*
Discrete Mathematics
*

We also obtain simple constructions of a

doi:10.1016/0012-365x(95)00104-5
fatcat:4btk53rlufaunc5x7hsugdonoi
*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. ...##
###
Total choosability of planar graphs with maximum degree 4

2011
*
Discrete Applied Mathematics
*

Let G be a

doi:10.1016/j.dam.2010.10.001
fatcat:k3nxloju6vdqlos3hosfu7tjhm
*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*. ...##
###
Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable—A short proof

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. ...

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