Choosability with separation of complete multipartite graphs and hypergraphs

On Choosability with Separation of Planar Graphs with Forbidden Cycles

“…The main problem if planar graphs are (3,1)-choosable remains open. We hope that this paper could serve as an inspiration of possible approaches to the problem.…”

“…(R3D) Otherwise, v gives charge 3 2 to each incident bad 3-face and distributes its remaining charge uniformly to each incident non-bad 3-face.…”

“…(R4) If d(v) ≥ 6, then v gives charge 1 to each incident normal 4-face and charge 3 2 to each incident 3-face.…”

On choosability with separation of planar graphs with lists of different sizes

“…For (1), note that |L(uv)| ≤ 1, and if L(uv) = ∅ then it suffices to L-color G − uv, which is possible by minimality. For (2), the definitions imply L(u) ⊇ v∈N (u) L(uv), and if γ ∈ L(u) \ v∈N (u) L(uv) then L-coloring G − u, and then coloring u with γ yields an Lcoloring of G. Finally consider (3). By (1), there exists a color γ with L(uv) = {γ} = L(vw).…”

“…They investigate this concept for both complete graphs and sparse graphs. The study of dense graphs were extended to complete bipartite graphs and multipartite graphs by Füredi, Kostochka, and Kumbhat [2,3].…”