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Title: Colouring graphs with forbidden bipartite subgraphs
Abstract A conjecture of Alon, Krivelevich and Sudakov states that, for any graph $$F$$ , there is a constant $$c_F \gt 0$$ such that if $$G$$ is an $$F$$ -free graph of maximum degree $$\Delta$$ , then $$\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$$ . Alon, Krivelevich and Sudakov verified this conjecture for a class of graphs $$F$$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot and Sereni that if $$G$$ is $$K_{t,t}$$ -free, then $$\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$$ as $$\Delta \to \infty$$ . We improve this bound to $$(1+o(1)) \Delta/\log\!\Delta$$ , making the constant factor independent of $$t$$ . We further extend our result to the DP-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle.  more » « less
Award ID(s):
2045412
PAR ID:
10418233
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Combinatorics, Probability and Computing
Volume:
32
Issue:
1
ISSN:
0963-5483
Page Range / eLocation ID:
45 to 67
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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