Algorithmic Aspects of Partial List Colourings

Abstract

Let G = (V , E) be a graph with n vertices, chromatic number χ(G) and list chromatic number χ (G). Suppose each vertex of V (G) is assigned a list of t colours. Albertson, Grossman and Haas [1] conjectured that at least t χ (G) n vertices can be coloured properly from these lists.Albertson, Grossman and Haas [1] and Chappell [3] proved partial results concerning this conjecture. This paper presents algorithms that colour at least the number of vertices given in the bounds of Albertson, Grossman and Haas, and Chappell. In particular, it follows that the conjecture is valid for all bipartite graphs and that, for every bipartite graph and every assignment of lists with t colours in each list where 0 6 t 6 χ (G), it is possible to colour at least (1 − (1/2) t )n vertices in polynomial time. Thus, if G is bipartite and L is a list assignment with |L(v)| > log 2 n for all v ∈ V , then G is L-list colourable in polynomial time.

Partial Online List Coloring of Graphs

“…The proof of Theorem 3 in uses the probabilistic method. In , Voigt gave an algorithmic proof of this result. The proof of Theorem 7 below uses a combination of ideas from and from .…”

“…In , Voigt gave an algorithmic proof of this result. The proof of Theorem 7 below uses a combination of ideas from and from . Lemma If and , then can be partitioned as such that and . Proof Assume the Lister and the Painter play the ‐list coloring game on G .…”

Partial DP-Coloring

“…Albertson, Grossman, and Haas [2] introduced partial list coloring with a "frankly mischievous" intent of inciting further work. Indeed, this has received attention in several papers [2,12,15,16,17,18,29]. Given a list assignment L, we want to properly L-color as many vertices as possible.…”

“…(see [2,29]), α ℓ t (G) > 6 7 (t|V (G)|/χ ℓ (G)) (see [12]), and [15,16]). The Partial List Coloring Conjecture has been proven for bipartite graphs (by the bound from [2,29]), graphs G with ∆(G) ≤ χ ℓ (G) (see [17]), claw-free graphs, chordless graphs, chordal graphs, series parallel graphs, and graphs G satisfying |V (G)| ≤ 2χ(G) + 1 (see [18]). Iradmusa [16] also showed that for every graph G, the inequality in Conjecture 1 holds for at least half the values of t in [χ ℓ (G) − 1].…”