Algorithmic complexity of proper labeling problems

Dehghan, Ali; Sadeghi, Mohammad‐Reza; Ahadi, Arash

doi:10.1016/j.tcs.2013.05.027

Cited by 37 publications

(49 citation statements)

References 25 publications

(32 reference statements)

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“…Thus, the first moment method yields the statement. 5. Almost all 5-uniform hypergraphs are either 1 or 2 strongly weighted…”

Section: Binomial Coefficients Approximationmentioning

confidence: 99%

“…2 ) k ) → λ k as n tends to infinity. Observe that (X (5) 2 ) k consists of n 2 n−2 2 · · · n−2k+2 2 terms of k vertex-disjoint pairs and O(n 2k−1 ) remaining terms. The pairs in the remaining terms are not vertex disjoint.…”

Section: Binomial Coefficients Approximationmentioning

confidence: 99%

“…Conjecture 1. 5. For every r ≥ 3 there is a constant w = w(r) such that each nice r-uniform hypergraph is strongly w-weighted.…”

Section: Introductionmentioning

confidence: 99%

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Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs

Bennett¹,

Dudek

Frieze³

et al. 2016

Electron. J. Combin.

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Given an r-uniform hypergraph H = (V, E) and a weight function ω : E → {1, . . . , w}, a coloring of vertices of H, induced by ω, is defined by c(v) = e v w(e) for all v ∈ V . If there exists such a coloring that is strong (that means in each edge no color appears more than once), then we say that H is strongly wweighted. Similarly, if the coloring is weak (that means there is no monochromatic edge), then we say that H is weakly w-weighted. In this paper, we show that almost all 3 or 4-uniform hypergraphs are strongly 2-weighted (but not 1-weighted) and almost all 5-uniform hypergraphs are either 1 or 2 strongly weighted (with a nontrivial distribution). Furthermore, for r ≥ 6 we show that almost all r-uniform hypergraphs are strongly 1-weighted. We complement these results by showing that almost all 3uniform hypergraphs are weakly 2-weighted but not 1-weighted and for r ≥ 4 almost all r-uniform hypergraphs are weakly 1-weighted. These results extend a previous work of Addario-Berry, Dalal and Reed for graphs. We also prove general lower bounds and show that there are r-uniform hypergraphs which are not strongly (r 2 − r)-weighted and not weakly 2-weighted. Finally, we show that determining whether a particular uniform hypergraph is strongly 2-weighted is NP-complete. arXiv:1511.04569v2 [math.CO]

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“…Thus, the first moment method yields the statement. 5. Almost all 5-uniform hypergraphs are either 1 or 2 strongly weighted…”

Section: Binomial Coefficients Approximationmentioning

confidence: 99%

Section: Binomial Coefficients Approximationmentioning

confidence: 99%

See 1 more Smart Citation

Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs

Bennett¹,

Dudek

Frieze³

et al. 2016

Electron. J. Combin.

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“…They introduced an edge-labeling which is additive vertex-coloring that means for every edge uv, the sum of labels of the edges incident to u is different from the sum of labels of the edges incident to v. It was conjectured in [8] that every graph with no isolated edge has a neighbour-sumdistinguishing labeling from N 3 (1-2-3-conjecture). This conjecture has been studied extensively by several authors, for instance see [2,6,8]. Currently, we know that every connected graph with more than two vertices has a neighbour-sum-distinguishing labeling, using the labels from N 5 [7].…”

Section: Introductionmentioning

confidence: 98%

Is there any polynomial upper bound for the universal labeling of graphs?

2016

Self Cite

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A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation ℓ of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from {1, 2, . . . , k} denoted it by G) , where ∆(G) denotes the maximum degree of G. In this work, we offer a provocative question that is:" Is there any polynomial function f such that for every graph G, − → χu(G) ≤ f (∆(G))?". Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree T , − → χu(T ) = O(∆ 3 ). Next, we show that for a given 3-regular graph G, the universal labeling number of G is 4 if and only if G belongs to Class 1. Therefore, for a given 3-regular graph G, it is an NP-complete to determine whether the universal labeling number of G is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem.

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“…[14]. For additional related labelings we refer to [8]. We mention in passing that graph labelings have a variety of applications such as incorporating redundancy in disks, designing drilling machines, creating layouts for circuit boards, and configuring resistor networks, see [18].…”

Section: Introductionmentioning

confidence: 99%

A lower bound and several exact results on the d-lucky number

Klavžar

Rajasingh

Emilet

2020

Applied Mathematics and Computation

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If ℓ : V (G) → N is a vertex labeling of a graph G = (V (G), E(G)), then the d-lucky sum of a vertex u ∈. A general lower bound on the d-lucky number of a graph in terms of its clique number and related degree invariants is proved. The bound is sharp as demonstrated with an infinite family of corona graphs. The d-lucky number is also determined for the so-called G n,m -web graphs and graphs obtained by attaching the same number of pendant vertices to the vertices of a generalized cocktail-party graph.

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Algorithmic complexity of proper labeling problems

Cited by 37 publications

References 25 publications

Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs

Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs

Is there any polynomial upper bound for the universal labeling of graphs?

A lower bound and several exact results on the d-lucky number

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