A refinement of choosability of graphs release_n47zikqlsbbd3oba5lazt6riaa

by Xuding Zhu

Released as a article .

2018  

Abstract

Assume k is a positive integer, λ={k_1, k_2, ..., k_q} is a partition of k and G is a graph. A λ-list assignment of G is a k-list assignment L of G such that the colour set ∪_v∈ V(G)L(v) can be partitioned into q subsets C_1 ∪ C_2 ...∪ C_q and for each vertex v of G, |L(v) ∩ C_i| > k_i. We say G is λ-choosable if for each λ-list assignment L of G, G is L-colourable. It follows from the definition that if λ ={k}, then λ-choosable is the same as k-choosable, if λ ={1,1,..., 1}, then λ-choosable is equivalent to k-colourable. For the other partitions of k sandwiched between {k} and {1,1,..., 1} in terms of refinements, λ-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions λ, λ' of k, every λ-choosable graph is λ'-choosable if and only if λ' is a refinement of λ. Then we concentrate on λ-choosability of planar graphs for partitions λ of 4. Several conjectures concerning colouring of generalized signed planar graphs are proposed and relations between these conjectures and list colouring conjectures for planar graphs are explored. 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 of Máčajová, Raspaud and Škoviera which asserts that every planar graph is signed MRS-4-colourable, and that a conjecture of Kang and Steffen asserting that every planar graph is signed KS-4-colourable implies that every planar graph is {1,1,2}-choosable.
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Date   2018-11-21
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arXiv  1811.08587v1
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