Extending Fractional Precolorings

Kráľ, Daniel; Krnc, Matjaž; Kupec, Martin; Lužar, Borut; Volec, Jan

doi:10.1137/110828988

Cited by 2 publications

(17 citation statements)

References 14 publications

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“…Finally, we state the following proposition which is implicit in the proof of Theorem 1.3 in [12]. Proposition 2.5 ( [12]). Let k = p/q be rational, where p, q ∈ N and p ≥ 2q, d, n ∈ N and ε > 0.…”

Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 94%

“…The graphs we introduce now are isomorphic to the ones defined in [12], although we use a slightly different notation.…”

Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 99%

“…An equivalent notion was used in [12] under the name universal product; the only difference is that the meaning of G and H was swapped, i.e., the universal product of G and H is isomorphic to the extension product of H and G. For a set X ∈…”

Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 99%

“…We start with the first two of them; the proof of the first one is straightforward and the proof of the second one is in [12]. The length of the shortest odd cycle of a graph G is the odd girth of G. The odd girth of the Kneser graph K p/q is equal to 2 q p − 2q +1, see [13].…”

Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 99%

“…The proof uses the same precoloring as was used in [12] for a lower bound in the case k ∈ {2} ∪ [3, ∞), but the argument for k ∈ (2, 3) is considerably more involved.…”

Section: Lower Bound For Distance Fourmentioning

confidence: 99%

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Extensions of Fractional Precolorings Show Discontinuous Behavior

Heuvel

Kráľ

Kupec

et al. 2014

Journal of Graph Theory

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We study the following problem: given a real number k and integer d, what is the smallest ε such that any fractional (k +ε)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k + ε)-coloring of the whole graph? The exact values of ε were known for k ∈ {2} ∪ [3, ∞) and any d. We determine the exact values of ε for k ∈ (2, 3) if d = 4, and k ∈ [2.5, 3) if d = 6, and give upper bounds for k ∈ (2, 3) if d = 5, 7, and k ∈ (2, 2.5) if d = 6. Surprisingly, ε viewed as a function of k is discontinuous for all those values of d.

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Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 94%

“…The graphs we introduce now are isomorphic to the ones defined in [12], although we use a slightly different notation.…”

Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 99%

Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 99%

Section: Notation Definitions and Preliminary Resultsmentioning

confidence: 99%

“…The proof uses the same precoloring as was used in [12] for a lower bound in the case k ∈ {2} ∪ [3, ∞), but the argument for k ∈ (2, 3) is considerably more involved.…”

Section: Lower Bound For Distance Fourmentioning

confidence: 99%

See 3 more Smart Citations

Extensions of Fractional Precolorings Show Discontinuous Behavior

Heuvel

Kráľ

Kupec

et al. 2014

Journal of Graph Theory

Self Cite

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Extending precolorings to distinguish group actions

Ferrara

Gethner

Hartke

et al. 2018

European Journal of Combinatorics

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Given a group Γ acting on a set X, a k-coloring φ : X → {1, . . . , k} of X is distinguishing with respect to Γ if the only γ ∈ Γ that fixes φ is the identity action. The distinguishing number of the action Γ, denoted D Γ (X), is then the smallest positive integer k such that there is a distinguishing k-coloring of X with respect to Γ. This notion has been studied in a number of settings, but by far the largest body of work has been concerned with finding the distinguishing number of the action of the automorphism group of a graph G upon its vertex set, which is referred to as the distinguishing number of G.The distinguishing number of a group action is a measure of how difficult it is to "break" all of the permutations arising from that action. In this paper, we aim to further differentiate the resilience of group actions with the same distinguishing number. In particular, we introduce a precoloring extension framework to address this issue. A set S ⊆ X is a fixing set for Γ if for every non-identity element γ ∈ Γ there is an element s ∈ S such that γ(s) = s. The distinguishing extension number ext D (X, Γ; k) is the minimum number m such that for all fixing sets W ⊆ X with |W | ≥ m, every k-coloring c : X \ W → [k] can be extended to a k-coloring that distinguishes X.In this paper, we prove that ext D (R, Aut(R), 2) = 4, where Aut(R) is comprised of compositions of translations and reflections. We also consider the distinguishing extension number of the circle and (finite) cycles, obtaining several exact results and bounds.

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Extending Fractional Precolorings

Cited by 2 publications

References 14 publications

Extensions of Fractional Precolorings Show Discontinuous Behavior

Extensions of Fractional Precolorings Show Discontinuous Behavior

Extending precolorings to distinguish group actions

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