<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mn>2</mml:mn></mml:math>-colorings in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-regular<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-uniform hypergraphs

Henning, Michael A.; Yeo, Anders

doi:10.1016/j.ejc.2013.04.005

Cited by 22 publications

(25 citation statements)

References 21 publications

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“…Proof Let e be any hyperedge containing v and let

H_{v}

be the subhypergraph with e removed from H . By the result of , the hypergraph

H_{v}

has a 2‐coloring, say

(A_{v}, B_{v})

. If we add v to whichever of

A_{v}

and

B_{v}

doesn’t already contain it, then both

A_{v}

and

B_{v}

are transversals of H .…”

Section: Cubic Graphsmentioning

confidence: 99%

“…In contrast, Thomassen [28] showed that, for ≥ 4, every -regular graph has two disjoint total dominating sets. It is known [17] that a connected 3-regular 3-uniform hypergraph is almost 2-colorable, in that there is a 2-coloring that 2-colors all but one specified hyperedge. Similarly, a connected cubic graph has a 2-coloring such that each vertex, except possibly two, has both colors in its neighborhood.…”

Section: Cubic Graphsmentioning

confidence: 99%

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Thoroughly dispersed colorings

Goddard

Henning

2017

Journal of Graph Theory

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We consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly dispersed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define td(G) as the maximum number of colors in such a coloring and FTD(G) as the fractional version thereof. In particular, we show that every claw‐free graph with minimum degree at least two has FTD(G)≥3/2 and this is best possible. For planar graphs, we show that every triangular disc has FTD(G)≥3/2 and this is best possible, and that every planar graph has td(G)≤4 and this is best possible, while we conjecture that every planar triangulation has td(G)≥2. Further, although there are arbitrarily large examples of connected, cubic graphs with td(G)=1, we show that for a connected cubic graph FTD(G)≥2−o(1). We also consider the related concepts in hypergraphs.

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“…Proof Let e be any hyperedge containing v and let

H_{v}

be the subhypergraph with e removed from H . By the result of , the hypergraph

H_{v}

has a 2‐coloring, say

(A_{v}, B_{v})

. If we add v to whichever of

A_{v}

and

B_{v}

doesn’t already contain it, then both

A_{v}

and

B_{v}

are transversals of H .…”

Section: Cubic Graphsmentioning

confidence: 99%

Section: Cubic Graphsmentioning

confidence: 99%

Thoroughly dispersed colorings

Goddard

Henning

2017

Journal of Graph Theory

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“…In this article, we study 2‐colorings in k ‐regular k ‐uniform hypergraphs. In the authors presented results on free sets in hypergraphs in

H_{k}

for large k . In particular, they showed that for any ε where

0 < ε < 1

and for sufficiently large k , every hypergraph

H \in {scriptH}_{k}

of order n has a free set of size at least

c_{k} \times n

, where

c_{k} = 1 - (prefixln (k) + {(prefixln (k))}^{2}) / ((1 - ε) k)

, and so

c_{k} \to 1

k \to \infty

.…”

Section: Resultsmentioning

confidence: 99%

“…We shall need the following lemma. Lemma () If

H \in {scriptH}_{k}

, then there exists a hypergraph

H^{'} \in {scriptH}_{k - 1}

on the same vertex set such that every edge of H contains an edge of

H^{'}

.…”

Section: Hypergraphs In Hk With Small Kmentioning

confidence: 99%

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On 2‐Colorings of Hypergraphs

Henning

Yeo

2014

Journal of Graph Theory

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In this article, we continue the study of 2‐colorings in hypergraphs. A hypergraph is 2‐colorable if there is a 2‐coloring of the vertices with no monochromatic hyperedge. Let Hk denote the class of all k‐uniform k‐regular hypergraphs. It is known (see Alon and Bregman [Graphs Combin. 4 (1988) 303–306] and Thomassen [J. Amer. Math. Soc. 5 (1992), 217–229] that every hypergraph H∈scriptHk is 2‐colorable, provided k≥4. As remarked by Alon and Bregman the result is not true when k=3, as may be seen by considering the Fano plane. Indeed there are several constructions for building infinite families of hypergraphs in H3 that are not 2‐colorable. Our main result in this paper is a strengthening of the above results. For this purpose, we define a set X of vertices in a hypergraph H to be a free set in H if we can 2‐color V(H)∖X such that every edge in H receives at least one vertex of each color. We conjecture that for k≥4, every hypergraph H∈scriptHk has a free set of size k−3 in H. We show that the bound k−3 cannot be improved for any k≥4 and we prove our conjecture when k∈{5,6,7,8}. Our proofs use results from areas such as transversal in hypergraphs, cycles in digraphs, and probabilistic arguments.

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Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph

Goddard

Henning

2020

Developments in Mathematics

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2-colorings ink-regulark-uniform hypergraphs

Cited by 22 publications

References 21 publications

Thoroughly dispersed colorings

Thoroughly dispersed colorings

On 2‐Colorings of Hypergraphs

Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph

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