A refinement of choosability of graphs

Abstract

Assume k is a positive integer, λ={k1,k2,…,kq} is a partition of k and G is a graph. A λ-assignment of G is a k-assignment L of G such that the colour set ⋃v∈V(G)L(v) can be partitioned into q subsets C1∪C2…∪Cq and for each vertex v of G, |L(v)∩Ci|=ki. We say G is λ-choosable if for each λ-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 study λ-choosability of special families of graphs. The Four Colour Theorem says that every planar graph is {1,1,1,1}-choosable. A very recent result of Kemnitz and Voigt implies that for any partition λ of 4 other than {1,1,1,1}, there is a planar graph which is not λ-choosable. We observe that, in contrast to the fact that there are non-4-choosable 3-chromatic planar graphs, every 3-chromatic planar graph is {1,3}-choosable, and that if G is a planar graph whose dual G⁎ has a connected spanning Eulerian subgraph, then G is {2,2}-choosable. We prove that if n is a positive even integer, λ is a partition of n−1 in which each part is at most 3, then Kn is edge λ-choosable. Finally we study relations between λ-choosability of graphs and colouring of signed graphs and generalized signed graphs. A conjecture of Máčajová, Raspaud and Škoviera that every planar graph is signed 4-colcourable is recently disproved by Kardoš and Narboni. We prove that every signed 4-colourable graph is weakly 4-choosable, and every signed Z4-colourable graph is {1,1,2}-choosable. The later result combined with the above result of Kemnitz and Voigt disproves a conjecture of Kang and Steffen that every planar graph is signed Z4-colourable. We shall show that a graph constructed by Wegner in 1973 is also a counterexample to Kang and Steffen's conjecture, and present a new construction of a non-{1,3}-choosable planar graphs.