Asymmetric List Sizes in Bipartite Graphs

Abstract Given a bipartite graph with parts A and B having maximum degrees at most $$\Delta _A$$ ? A _B$$ B , respectively, consider list assignment such that every vertex in or is given of colours size $$k_A$$ k $$k_B$$ respectively. We prove some general sufficient conditions terms to be guaranteed proper colouring each coloured using only colour from its list. These are asymptotically nearly sharp the very asymmetric cases. establish one condition particular, where _A=\Delta _B=\Delta $$ = $$k_A=\log \Delta log $$k_B=(1+o(1))\Delta /\log ( 1 + o ) / as \rightarrow \infty ? ? . This amounts partial progress towards conjecture 1998 Krivelevich first author. also derive necessary through an intriguing connection between complete case extremal approximate Steiner systems. show for graphs these large part parameter space. has provoked following. In setup above, we always if $$k_A \ge _A^\varepsilon ? ? $$k_B _B^\varepsilon any $$\varepsilon >0$$ > 0 provided enough; C \log C absolute constant $$C>1$$ ; _B = k_B (\Delta )^{1/k_A}\log $$C>0$$ generalisations above-mentioned author, true close best possible. Our provide conjectures.