Abstract
Given a bipartite graph with parts A and B having maximum degrees at most Delta _A and Delta _B, respectively, consider a list assignment such that every vertex in A or B is given a list of colours of size k_A or k_B, respectively. We prove some general sufficient conditions in terms of Delta _A, Delta _B, k_A, k_B to be guaranteed a proper colouring such that each vertex is coloured using only a colour from its list. These are asymptotically nearly sharp in the very asymmetric cases. We establish one sufficient condition in particular, where Delta _A=Delta _B=Delta , k_A=log Delta and k_B=(1+o(1))Delta /log Delta as Delta rightarrow infty . This amounts to partial progress towards a conjecture from 1998 of Krivelevich and the first author. We also derive some necessary conditions through an intriguing connection between the complete case and the extremal size of approximate Steiner systems. We show that for complete bipartite graphs these conditions are asymptotically nearly sharp in a large part of the parameter space. This has provoked the following. In the setup above, we conjecture that a proper list colouring is always guaranteedif k_A ge Delta _A^varepsilon and k_B ge Delta _B^varepsilon for any varepsilon >0 provided Delta _A and Delta _B are large enough;if k_A ge C log Delta _B and k_B ge C log Delta _A for some absolute constant C>1; orif Delta _A=Delta _B = Delta and k_B ge C (Delta /log Delta )^{1/k_A}log Delta for some absolute constant C>0. These are asymmetric generalisations of the above-mentioned conjecture of Krivelevich and the first author, and if true are close to best possible. Our general sufficient conditions provide partial progress towards these conjectures.
Highlights
Whereby arbitrary restrictions on the possible colours used per vertex are imposed, was introduced independently by Erdos, Rubin and Taylor [10] and by Vizing [16]
Note though that these last two examples show that Theorem 6 provides suboptimal necessary conditions for-choosability even in the complete case
We have begun the investigation of an asymmetric form of list colouring for bipartite graphs
Summary
Whereby arbitrary restrictions on the possible colours used per vertex are imposed, was introduced independently by Erdos, Rubin and Taylor [10] and by Vizing [16]. The complete bipartite graph G = (V =A∪B, E) with |A| = M (k1, kA, ) and |B| = M (k2, kB, ) is not (kA, kB)-choosable This link allows us, using known results for M , to read off decent necessary conditions for specific parameterisations of Problem 3. Note though that these last two examples show that Theorem 6 provides suboptimal necessary conditions for (kA, kB)-choosability even in the complete case. The complete bipartite graphs demonstrate its hypothetical sharpness up to a constant factor for the entire range of possibilities for kA, and symmetrically kB (Theorem 16(iii)). Theorem 4 provides the following partial progress towards Conjecture 7 under condition (iii) This constitutes significant (asymmetric) progress towards Conjecture 2, and is a first concrete step in this longstanding problem. If ep(d + 1) ≤ 1, with positive probability none of the events in E occur
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