A refinement of choosability of graphs
release_n47zikqlsbbd3oba5lazt6riaa
by
Xuding Zhu
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.
In text/plain
format
Archived Files and Locations
application/pdf 426.4 kB
file_tgsnlg6p4zbdbpadc64xjiv7va
|
arxiv.org (repository) web.archive.org (webarchive) |
1811.08587v1
access all versions, variants, and formats of this works (eg, pre-prints)