Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

Kwaśny, Jakub; Przybyło, Jakub

doi:10.1002/rsa.20813

Cited by 6 publications

(3 citation statements)

References 37 publications

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“…The least number of colours admitting an inclusion‐free colouring of

G

is called its inclusion chromatic index and denoted by

χ'_{\subset} (G)

. This problem was proposed by Simonyi [16], inspired by the concept discussed in the following paragraph (and in particular its list analogue, see [12]).…”

Section: Introductionmentioning

confidence: 99%

On the inclusion chromatic index of a graph

Przybyło

Kwaśny

2020

Journal of Graph Theory

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Let χ′⊂(G) be the least number of colours necessary to properly colour the edges of a graph G with minimum degree δgoodbreakinfix≥2 so that the set of colours incident with any vertex is not contained in a set of colours incident to any of its neighbours. We provide an infinite family of examples of graphs G with χ′⊂MathClass-open(GMathClass-close)≥true(1+1δ−1true)Δ, where normalΔ is the maximum degree of G, and we conjecture that χ′⊂ MathClass-open( G MathClass-close) ≤ true⌈ true( 1 + 1 δ − 1 true) Δ true⌉ for every connected graph with δ goodbreakinfix≥ 2 which is not isomorphic to C 5. The equality here is attained, for example, for the family of complete bipartite graphs. Using a probabilistic argument we support this conjecture by proving that χ ′ ⊂ MathClass-open( G MathClass-close) ≤ true( 1 + 4 δ true) Δ MathClass-open( 1 + o MathClass-open( 1 MathClass-close) MathClass-close) (for normalΔ goodbreakinfix→ ∞) for any fixed δ goodbreakinfix≥ 2, which implies that χ ′ ⊂ MathClass-open( G MathClass-close) ≤ true( 1 + 4 δ − 1 true) Δ for normalΔ large enough.

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“…The least number of colours admitting an inclusion‐free colouring of

G

is called its inclusion chromatic index and denoted by

χ'_{\subset} (G)

. This problem was proposed by Simonyi [16], inspired by the concept discussed in the following paragraph (and in particular its list analogue, see [12]).…”

Section: Introductionmentioning

confidence: 99%

On the inclusion chromatic index of a graph

Przybyło

Kwaśny

2020

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…The least number of colours admitting an inclusion-free colouring of G is called its inclusion chromatic index and denoted by χ ′ ⊂ (G). This problem was proposed by Simonyi [16], inspired by the concept discussed in the following paragraph (and in particular its list analogue, see [12]).…”

Section: Introductionmentioning

confidence: 99%

On inclusion chromatic index of a graph

Kwaśny¹,

Przybyło²

2019

Preprint

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Let χ ′⊂ (G) be the least number of colours necessary to properly colour the edges of a graph G with minimum degree δ ≥ 2 so that the set of colours incident with any vertex is not contained in a set of colours incident to any its neighbour. We provide an infinite family of examples of graphs G with χ ′ ⊂ (G) ≥ (1 + 1 δ−1 )∆, where ∆ is the maximum degree of G, and we conjecture that χ ′ ⊂ (G) ≤ ⌈(1 + 1 δ−1 )∆⌉ for every connected graph with δ ≥ 2 which is not isomorphic to C 5 . The equality here is attained e.g. for the family of complete bipartite graphs. Using a probabilistic argument we support this conjecture by proving that for any fixed δ, what implies that χ ′ ⊂ (G) ≤ (1 + 4 δ−1 )∆ for ∆ large enough.

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“…For example, one can ask for an edge-labeling, which distinguishes neighbors by sums/multisetes/sums, and is also required to be proper: for distinguishing neighbors by sums of labels, see e.g. Przybyło [47,48], Bonamy and Przybyło [14], Hocquard and Przybyło [33]; for distinguishing neighbors by sets of labels, see Zhang, Liu, and Wang [65], Balister, Győri, Lehel, and Schelp [6], Edwards, Horňák, and Woźniak [22], Bonamy, Bousquet, and Hocquard [13] or Hatami [31]; for list edge labelings distinguishing neighbors by multisets see a recent exciting result by Kwaśny and Przybyło [40].…”

Section: Introductionmentioning

confidence: 99%

Intersecting edge distinguishing colorings of hypergraphs

Okrasa¹,

Rzążewski²

2018

Preprint

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An edge labeling of a graph distinguishes neighbors by sets (multisets, resp.), if for any two adjacent vertices u and v the sets (multisets, resp.) of labels appearing on edges incident to u and v are different. In an analogous way we define total labelings distinguishing neighbors by sets or multisets: for each vertex, we consider labels on incident edges and the label of the vertex itself.In this paper we show that these problems, and also other problems of similar flavor, admit an elegant and natural generalization as a hypergraph coloring problem. An ieds-coloring (iedmcoloring, resp.) of a hypergraph is a vertex coloring, in which the sets (multisets, resp.) of colors, that appear on every pair of intersecting edges are different. We show upper bounds on the size of lists, which guarantee the existence of an ieds-or iedm-coloring, respecting these lists. The proof is essentially a randomized algorithm, whose expected time complexity is polynomial. As corollaries, we derive new results concerning the list variants of graph labeling problems, distinguishing neighbors by sets or multisets. We also show that our method is robust and can be easily extended for different, related problems.We also investigate a close connection between edge labelings of bipartite graphs, distinguishing neighbors by sets, and the so-called property B of hypergraphs. We discuss computational aspects of the problem and present some classes of bipartite graphs, which admit such a labeling using two labels.

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Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

Cited by 6 publications

References 37 publications

On the inclusion chromatic index of a graph

On the inclusion chromatic index of a graph

On inclusion chromatic index of a graph

Intersecting edge distinguishing colorings of hypergraphs

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