TY - JOUR
T1 - Asymmetric List Sizes in Bipartite Graphs
AU - Alon, Noga
AU - Cambie, Stijn
AU - Kang, Ross J.
N1 - Publisher Copyright:
© 2021, The Author(s).
PY - 2021/12
Y1 - 2021/12
N2 - Given a bipartite graph with parts A and B having maximum degrees at most Δ A and Δ B, respectively, consider a list assignment such that every vertex in A or B is given a list of colours of size kA or kB, respectively. We prove some general sufficient conditions in terms of Δ A, Δ B, kA, kB 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 Δ A= Δ B= Δ , kA= log Δ and kB= (1 + o(1)) Δ / log Δ as Δ → ∞. 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 kA≥ΔAε and kB≥ΔBε for any ε> 0 provided Δ A and Δ B are large enough;if kA≥ Clog Δ B and kB≥ Clog Δ A for some absolute constant C> 1 ; orif Δ A= Δ B= Δ and kB≥C(Δ/logΔ)1/kAlogΔ 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.
AB - Given a bipartite graph with parts A and B having maximum degrees at most Δ A and Δ B, respectively, consider a list assignment such that every vertex in A or B is given a list of colours of size kA or kB, respectively. We prove some general sufficient conditions in terms of Δ A, Δ B, kA, kB 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 Δ A= Δ B= Δ , kA= log Δ and kB= (1 + o(1)) Δ / log Δ as Δ → ∞. 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 kA≥ΔAε and kB≥ΔBε for any ε> 0 provided Δ A and Δ B are large enough;if kA≥ Clog Δ B and kB≥ Clog Δ A for some absolute constant C> 1 ; orif Δ A= Δ B= Δ and kB≥C(Δ/logΔ)1/kAlogΔ 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.
UR - http://www.scopus.com/inward/record.url?scp=85114015498&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85114015498&partnerID=8YFLogxK
U2 - 10.1007/s00026-021-00552-5
DO - 10.1007/s00026-021-00552-5
M3 - Article
AN - SCOPUS:85114015498
SN - 0218-0006
VL - 25
SP - 913
EP - 933
JO - Annals of Combinatorics
JF - Annals of Combinatorics
IS - 4
ER -