A lower bound for partial list colorings

Chappell, Glenn G.

doi:10.1002/(sici)1097-0118(199912)32:4<390::aid-jgt6>3.0.co;2-d

Cited by 7 publications

(6 citation statements)

References 2 publications

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“…Hence the number of colored vertices is at least

(1 - p) n

. It was shown in that

(1 - p) > \frac{6}{7} \frac{t}{r + t}

. Therefore,

λ_{t}^{OL} (G) > \frac{6}{7} \frac{t n}{s}

.…”

Section: The Proofsmentioning

confidence: 90%

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Partial Online List Coloring of Graphs

Wong

Zhu

2013

Journal of Graph Theory

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For a graph G, let λ t (G) be the maximum number of vertices of G that can be colored whenever each vertex of G is given t permissible colors. Albertson, Grossman, and Haas conjectured that if G is s-choosable and t ≤ s, then λ t ≥ t s |V (G)|. In this article, we consider the online version of this conjecture. Let λ OL t (G) be the maximum number of vertices of G that can be colored online whenever each vertex of G is given t permissible colors online. An analog of the above conjecture is the following: if G is online s-choosable and t ≤ s, then λ OL t (G) ≥ t s |V (G)|. This article generalizes some results concerning partial list coloring to online partial list coloring. We prove that for any positive integers p, q, λ OL p (G) + λ OL q (G) ≥

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“…Hence the number of colored vertices is at least

(1 - p) n

. It was shown in that

(1 - p) > \frac{6}{7} \frac{t}{r + t}

. Therefore,

λ_{t}^{OL} (G) > \frac{6}{7} \frac{t n}{s}

.…”

Section: The Proofsmentioning

confidence: 90%

“…The proof of Theorem 3 in uses the probabilistic method. In , Voigt gave an algorithmic proof of this result.…”

Section: The Proofsmentioning

confidence: 99%

Partial Online List Coloring of Graphs

Wong

Zhu

2013

Journal of Graph Theory

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“…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.…”

Section: Partial List Coloringmentioning

confidence: 99%

Partial DP-Coloring

Kaul,

Mudrock,

Pelsmajer

2020

Preprint

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In 1980 Albertson and Berman introduced partial coloring and then in 2000, Albertson, Grossman, and Haas introduced partial list coloring. Here we initiate the study of partial coloring for DP-coloring (aka, correspondence coloring), a recent insightful generalization of list coloring introduced in 2015 by Dvořák and Postle. The partial t-chromatic number of a graph G, denoted α t (G), is the maximum number of vertices that can be colored with t colors. Clearly,be the maximum number of vertices that can be colored from those lists. The partial t-choice number of a graph G, denoted α ℓ t (G), is the minimum of α L (G) taken over all assignments L for which |L(v)| = t for each v ∈ V (G). The Partial List Coloring Conjecture states that for any graph G, α ℓ t (G) ≥ t|V (G)|/χ ℓ (G) whenever t ∈ {1, . . . , χ ℓ (G)} where χ ℓ (G) is the list chromatic number of G. We show that while the DP-coloring analogue of the Partial List Coloring Conjecture does not hold, several results on partial list coloring can be extended to the DP-coloring context. We also study partial DP-coloring of the join of a graph with a complete graph, and we present several interesting open questions.

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“…Albertson, Grossman and Haas [1] proved that λ t > (1 − (1 − 1 χ ) t )n for all t between 0 and χ (G) and that this number is asymptotically best possible. Furthermore, Chappell [3] gave the lower bound 6 7 t χ (G) n for λ t for all t with 0 6 t 6 χ (G).…”

Section: Conjecture 11 (Albertson Grossman and Haasmentioning

confidence: 99%

Algorithmic Aspects of Partial List Colourings

Voigt

2000

Combinator. Probab. Comp.

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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.

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A lower bound for partial list colorings

Cited by 7 publications

References 2 publications

Partial Online List Coloring of Graphs

Partial Online List Coloring of Graphs

Partial DP-Coloring

Algorithmic Aspects of Partial List Colourings

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