You Can't Paint Yourself into a Corner

Albertson, Michael O.

doi:10.1006/jctb.1998.1827

Cited by 54 publications

(102 citation statements)

References 12 publications

Supporting

Mentioning

101

Contrasting

Order By: Relevance

“…Shortly after that, the problem was answered in affirmative by Albertson [1] who established the following theorem.…”

Section: Problemmentioning

confidence: 99%

See 1 more Smart Citation

Extending Fractional Precolorings

Kráľ¹,

Krnc²,

Kupec³

et al. 2012

SIAM J. Discrete Math.

View full text Add to dashboard Cite

For every d ≥ 3 and k ∈ {2} ∪ [3, ∞), we determine the smallest ε such that every fractional (k + ε)-precoloring of vertices at mutual distance at least d of a graph G with fractional chromatic number equal to k can be extended to a proper fractional (k + ε)-coloring of G. Our work complements the analogous results of Albertson for ordinary colorings and those of Albertson and West for circular colorings.

show abstract

“…Shortly after that, the problem was answered in affirmative by Albertson [1] who established the following theorem.…”

Section: Problemmentioning

confidence: 99%

“…It is possible to show that the constraint on the mutual distance between the vertices of W to be at least four in Theorem 2 is the best possible even if the graph G is planar, see [1] for further details.…”

Section: Problemmentioning

confidence: 99%

Extending Fractional Precolorings

Kráľ¹,

Krnc²,

Kupec³

et al. 2012

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…The following question was asked by Albertson [2] (see also Thomassen [28] Albertson himself [2] answered this question for the usual graph coloring case. In fact, his theorem says that the distance 4 is enough instead of 100.…”

Section: Inputmentioning

confidence: 99%

List-Color-Critical Graphs on a Fixed Surface

Kawarabayashi¹,

Mohar²

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

A k-list-assignment for a graph G assigns to each vertex v of G a list L(v) of admissible colors, where |L(v)| ≥ k. A graph is k-list-colorable (or k-choosable) if it can be properly colored from the lists for every k-listassignment.We prove the following conjecture posed by Thomassen in 1994: "There are only finitely many listcolor-critical graphs with all lists of cardinality at least 5 on any fixed surface." This generalizes the well-known result of Thomassen on the usual graph coloring case. We use this theorem and specific parts of its proof to resolve the complexity status of the following problem about k-list-coloring graphs on a fixed surface S, where k is a fixed positive integer. The cases k = 3, 4 are known to be NP-hard (actually even Π p 2 -complete), and the cases k = 1, 2 are easy. Our main results imply that the problem is tractable for every k ≥ 5. In fact, together with our recent algorithmic result, we are able to solve it in linear time when k ≥ 5. Our proof yields even more: if the input graph is k-list-colorable, then for any k-listassignment L, we can construct an L-coloring of G in linear time. This generalizes the well-known linear-time algorithms for planar graphs by Nishizeki and Chiba (for 5-coloring), and Thomassen (for 5-list-coloring).We also give a polynomial-time algorithm to resolve the following question: If the graph G is k-list-colorable, then our first result gives a linear time solution. However, the second problem is more general, since it provides a coloring (or a small obstruction) for an arbitrary graph in S.We also use our main theorem to prove another conjecture that was proposed recently by Thomassen: "For every fixed surface S, there exists a positive constant c such that every 5-list-colorable graph with n vertices embedded on S, has at least c·2 n distinct 5-listcolorings for every 5-list-assignment for G." Thomassen himself proved that this conjecture holds for usual 5-colorings.In addition to all these results, we also made partial progress towards a conjecture of Albertson concerning coloring extensions and a progress on similar questions for triangle-free graphs and graphs of larger girth.

show abstract

“…The proof uses an amalgamation of ideas from [1,3,7,17,31]. The overall structure is: First, given G we construct a convenient planar graph G 0 .…”

Section: A 5-color Extension Theoremmentioning

confidence: 99%

“…It turns out that the answer does not depend on an embedding. Theorem 1 [1]. If 1G r and the distance between any two vertices in P is at least 4, then any r 1-coloring of P extends to an r 1-coloring of all of G.…”

Section: Introductionmentioning

confidence: 96%