A new type of edge-derived vertex coloring

Gyri, Ervin; Palmer, Cory

doi:10.1016/j.disc.2008.11.014

Cited by 11 publications

(12 citation statements)

References 8 publications

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“…(In [3] there are classified bipartite graphs with respect to the general neighbour-distinguishing index.) Provided k = 2, the first upper bound for gndi(G) given by [4] is 3 1 2 log 2 4 = 3 = k + 1 so that the upper bound of Theorem 2 is not attained. This leads to the following natural question: Problem 1.…”

Section: Now Consider the Setmentioning

confidence: 95%

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General neighbour-distinguishing index via chromatic number

Horňák

Soták

2010

Discrete Mathematics

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Section: Now Consider the Setmentioning

confidence: 95%

“…Győri and Palmer in [4] improved the upper bound for the general neighbour-distinguishing index of G to 3 1 2 log 2 χ (G) . They have also proved that if χ (G) ≥ 5, then gndi(G) ≤ log 2 (χ (G) − 3) + 5.…”

mentioning

confidence: 97%

General neighbour-distinguishing index via chromatic number

Horňák

Soták

2010

Discrete Mathematics

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“…For more information we refer the reader to a survey on the 1-2-3 Conjecture and related problems by Seamone [25] (also see [4,6,8,10,12,24,30]). Different variations of distinguishing labelings of graphs have also been considered, see [5,7,17,20,21,22,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

On the algorithmic complexity of adjacent vertex closed distinguishing colorings number of graphs

Dehghan

Mollahajiaghaei

2017

Discrete Applied Mathematics

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An assignment of numbers to the vertices of graph G is closed distinguishing if for any two adjacent vertices v and u the sum of labels of the vertices in the closed neighborhood of the vertex v differs from the sum of labels of the vertices in the closed neighborhood of the vertex u unless they have the same closed neighborhood (i.e. N [u] = N [v]). The closed distinguishing number of a graph G, denoted by dis [G], is the smallest integer k such that there is a closed distinguishing labeling for G using integers from the set {1, 2, . . . , k}. Also, for each vertex v ∈ V (G), let L(v) denote a list of natural numbers available at v. A list closed distinguishing labeling is a closed distinguishing labeling f such that f (v) ∈ L(v) for each v ∈ V (G). A graph G is said to be closed distinguishing k-choosable if every k-list assignment of natural numbers to the vertices of G permits a list closed distinguishing labeling of G. The closed distinguishing choice number of G, dis ℓ [G], is the minimum natural number k such that G is closed distinguishing k-choosable. In this work we show that for each integer t there is a bipartite graph G such that dis[G] > t. This is an answer to a question raised by Axenovich et al. in [3] that how "dis" function depends on the chromatic number of a graph. It was shown that for every graph G with ∆ ≥ 2, dis[G] ≤ dis ℓ [G] ≤ ∆ 2 − ∆ + 1 and also there are infinitely many values of ∆ for which G might be chosen so that dis[G] = ∆ 2 − ∆ + 1 [3]. In this work, we prove that the difference between dis[G] and dis ℓ [G] can be arbitrary large and show that for every positive integer t there is a graph G such that dis ℓ [G] − dis[G] ≥ t. Also, we improve the current upper bound and give some number of upper bounds for the closed distinguishing choice number by using the Combinatorial Nullstellensatz. Among other results, we show that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. Also, we prove that for every k ≥ 3, it is NP-complete to decide whether dis[G] = k for a given graph G. * E-mail addresses: alidehghan@sce.carleton.ca, mmollaha@uwo.ca.

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“…For instance, one may vary the set to be weighted or coloured (edges, vertices, or both), the operation by which one obtains a colour, or the set from which one weights the objects (a variation of interest is to study weightings from arbitrary lists rather than from a fixed set). Due to the number of different weighting-colouring parameters one could conceivably study (and since there is no standard notation that is consistent across the literature on the subject), we have introduced the notation above, which is a modification of notation proposed by Győri and Palmer [50]. Note the three components of χ e Σ (G) -"χ" indicates that we desire a proper vertex colouring, the superscript "e" indicates that the weighting of interest is of the edges, and the subscript "Σ" indicates that vertex colours are obtained by summing edge weights.…”

Section: Introductionmentioning

confidence: 99%

The 1-2-3 Conjecture and related problems: a survey

Seamone¹

2012

Preprint

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The 1-2-3 Conjecture, posed in 2004 by Karoński, Luczak, and Thomason, states that one may weight the edges of any connected graph on at least 3 vertices from the set {1, 2, 3} (call the weight function w) so that the function f (v) = u∈N (v) w(uv) is a proper vertex colouring. This paper presents the current state of research on the 1-2-3 Conjecture and the many variants that have been proposed in its short but active history.

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A new type of edge-derived vertex coloring

Abstract: a b s t r a c tWe study the minimum number of weights assigned to the edges of a graph G with no component K 2 so that any two adjacent vertices have distinct sets of weights on their incident edges. The best possible upper bound on this parameter is proved.

Cited by 11 publications

References 8 publications

General neighbour-distinguishing index via chromatic number

General neighbour-distinguishing index via chromatic number

On the algorithmic complexity of adjacent vertex closed distinguishing colorings number of graphs

The 1-2-3 Conjecture and related problems: a survey

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