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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.  ...

### A not 3-choosable planar graph without 3-cycles

Margit Voigt
1995 Discrete Mathematics

### 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.  ...

### 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  ...

### 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  ...

### 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.  ...

### 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.  ...

### 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.  ...
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