A note on improper DP-colouring of planar graphs. Issue 2 (3rd April 2021) - Cai, Hongyan; Sun, Qiang
ABSTRACT: DP-colouring (also known as correspondence colouring), introduced by Dvořák and Postle, is a generalization of list colouring. Many results on list-colouring of graphs, especially of planar graphs, have been extended to the setting of DP-colouring. Recently, Pongpat and Kittikorn [P. Sittitrai and K. Nakprasit, Suffficient conditions on planar graphs to have a relaxed DP-3-colourability, Graphs and Combinatorics 35 (2019), pp. 837–845.] introduced DP- ( k, d ) ∗ -colouring to generalize ( k, d ) ∗ -colouring and ( k, d ) ∗ -choosability. They proved that every planar graph G without 4, 6 -cycles is DP- ( 3, 1 ) ∗ -colourable. In this note, we show the following results:(1) Every planar graph G without 4, 5, 7 -cycles is DP- ( 3, 1 ) ∗ -colourable; (2) Every planar graph G without 4, 5, 9 -cycles is DP- ( 3, 1 ) ∗ -colourable; (3) Every planar graph G without 4, 8 -cycles is DP- ( 3, 1 ) ∗ -colourable.