List Colouring Squares of Planar Graphs

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

“…However, asymptotically we can do much better. Havet, van den Heuvel, McDiarmid, and Reed [81] proved that Wegner's Planar Graph Conjecture is true asymptotically, even for list coloring.…”

“…But in a sense, the key idea is that list coloring squares of planar graphs is very similar to list coloring line graphs. This problem was solved asymptotically by Kahn [102], and the proof of the key lemma in [81] follows the approach in [102]. This approach was extended by Amini, Esperet, and van den Heuvel [7] to the case of graphs embeddable in any fixed surface.…”

“…More precisely, for any fixed surface S, Theorem 3.2 remains true when we replace "planar graph" with "graph embeddable in S". In [81] the authors also posed the following more general conjecture.…”

$2$-distance, injective, and exact square list-coloring of planar graphs with maximum degree 4

“…In 2018, the case when ∆ ≤ 3 was proved independently by Thomassen [42] and by Hartke et al [25]. Additionally, for ∆ ≥ 8, Havet et al [27] proved that the bound is 3 2 ∆(1 + o( 1)). Moreover, Conjecture 1 is known to be true for some subfamilies of planar graphs (e.g., K 4 -minor free graphs [36]).…”

“…Thus, a p-distance coloring is a generalization of the the classical proper coloring. Throughout the years, p-distance colorings, in particular the case when p = 2, became a focus for many researchers, see [2,26,27,42]. Only in recent years, several results were investigated with regards to the 2-distance coloring of planar graphs, see [4,13,20].…”