Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture

Kalkowski, Maciej; Karoński, Michał; Pfender, Florian

doi:10.1016/j.jctb.2009.06.002

Cited by 218 publications

(128 citation statements)

References 6 publications

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“…Karoński, Łuczak and Thomason in [8] considered general colorings of edges, and they conjectured that three colors are enough to distinguish adjacent vertices by sums. This conjecture is almost proved-Kalkowski showed that five colors are enough [7]. A similar conjecture by Przybyło and Woźniak states that two colors are enough by general total coloring [9].…”

Section: Conjecure 1 For Every Graph G = (V E) the Total-neighbor-dmentioning

confidence: 96%

On the Total-Neighbor-Distinguishing Index by Sums

Pilśniak

Woźniak

2013

Graphs and Combinatorics

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We consider a proper coloring c of edges and vertices in a simple graph and the sum f (v) of colors of all the edges incident to v and the color of a vertex v. We say that a coloring c distinguishes adjacent vertices by sums, if every two adjacent vertices have different values of f . We conjecture that + 3 colors suffice to distinguish adjacent vertices in any simple graph. In this paper we show that this holds for complete graphs, cycles, bipartite graphs, cubic graphs and graphs with maximum degree at most three.

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Section: Conjecure 1 For Every Graph G = (V E) the Total-neighbor-dmentioning

confidence: 96%

On the Total-Neighbor-Distinguishing Index by Sums

Pilśniak

Woźniak

2013

Graphs and Combinatorics

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“…A k -weighting is additive vertex-coloring if for every edge uv, the sum of weights of the edges incident to u is different than the sum of weights of the edges incident to v. They conjectured that every non-trivial graph permits an additive vertex-coloring 3-weighting (1-2-3 Conjecture), and proved this conjecture for 3-colorable graphs. The best result concerning 1-2-3 Conjecture is given by Kalkowski et al (2010), who presented an additive vertex-coloring 5-weighting for every nontrivial graph.…”

Section: K-weighting Of a Graph G Is Called Multiplicative Vertex-colmentioning

confidence: 99%

“…Different versions of vertex coloring from an edge k-weighting (by considering the sums, products, sequences, sets, or multisets of incident edge weights) were investigated by many authors in Addario-Berry et al (2005), Bartnicki et al (2009), Chang et al (2011, Kalkowski et al (2010), Karoński et al (2004), Lu et al (2011), Skowronek-Kaziów (2012, Stevens and Seamone (2013). Some authors distinguish all the vertices in a graph by their product colors (product irregularity strength of graphs, see Anholcer (2009Anholcer ( , 2014, Darda and Hujdurovic (2014)).…”

Section: K-weighting Of a Graph G Is Called Multiplicative Vertex-colmentioning

confidence: 99%

Graphs with multiplicative vertex-coloring 2-edge-weightings

Skowronek-Kaziów

2015

J Comb Optim

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A k-weighting w of a graph is an assignment of an integer weight w(e) ∈ {1, ...k} to each edge e. Such an edge weighting induces a vertex coloring c defined by c(v) = v∈e w(e). A k-weighting of a graph G is multiplicative vertex-coloring if the induced coloring c is proper, i.e., c(u) = c(v) for any edge uv ∈ E(G). This paper studies the parameter μ(G), which is the minimum k for which G has a multiplicative vertex-coloring k-weighting. Chang, Lu, Wu, Yu investigated graphs with additive vertex-coloring 2-weightings (they considered sums instead of products of incident edge weights). In particular, they proved that 3-connected bipartite graphs, bipartite graphs with the minimum degree 1, and r -regular bipartite graphs with r ≥ 3 permit an additive vertex-coloring 2-weighting. In this paper, the multiplicative version of the problem is considered. It was shown in Skowronek-Kaziów (Inf Process Lett 112:191-194, 2012) that μ(G) ≤ 4 for every graph G. It was also proved that every 3-colorable graph admits a multiplicative vertex-coloring 3 -weighting. A natural problem to consider is whether every 2-colorable graph (i.e., a bipartite graph) has a multiplicative vertex-coloring 2-weighting. But the answer is no, since the cycle C 6 and the path P 6 do not admit a multiplicative vertex-coloring 2-weighting. The paper presents several classes of 2-colorable graphs for which μ(G) = 2, including trees with at least two adjacent leaf edges, bipartite graphs with the minimum degree 3 and bipartite graphs G = (A, B, E) with even |A| or |B|.

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“…. , 16}, and recently, it is shown in [10] that the conjecture would hold if the set is {1, 2, 3, 4, 5}.…”

Section: Introductionmentioning

confidence: 98%

“…. , 11} for all y ∈ V ∪ E. This result was improved in [10] where it was shown f can be chosen so that f (v) ∈ {1, 2} for every vertex v and f (e) ∈ {1, 2, 3} for every edge e.…”

Section: Introductionmentioning

confidence: 99%

List Total Weighting of Graphs

Wong

Zhu

Yang

2010

Fete of Combinatorics and Computer Science

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A graph G = (V, E) is (k, k )-total weight choosable if the following is true: For any (k, k )-total list assignment L that assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k real numbers as permissible weights, there is a proper L-total weighting, i.e., a mapping f : V ∪ E → R such that f (y) ∈ L(y) for each y ∈ V ∪ E, and for any two adjacent vertices u and v, e∈E(u) (v). This paper introduces a method, the max-min weighting method, for finding proper L-total weightings of graphs. Using this method, we prove that complete multipartite graphs of the form K n,m,1,1,...,1 are (2, 2)-total weight choosable and complete bipartite graphs other than K 2 are (1, 2)-total weight choosable.

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Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture

Cited by 218 publications

References 6 publications

On the Total-Neighbor-Distinguishing Index by Sums

On the Total-Neighbor-Distinguishing Index by Sums

Graphs with multiplicative vertex-coloring 2-edge-weightings

List Total Weighting of Graphs

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