Weight choosability of graphs

Bartnicki, Tomasz; Grytczuk, Jarosław; Niwczyk, Stanisław

doi:10.1002/jgt.20354

Cited by 74 publications

(101 citation statements)

References 12 publications

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“…To be precise, the following stronger conjectures are proposed in [7,13] In [13], we had limited success and proved that if G = K 2,n , then G is (1, 2)-choosable, and if G is a complete graph, or G = K 3,n , or G is a tree or a generalized theta graph, then Conjecture 4.1 holds for G, and hence G is (2, 2)-choosable. We were unable to prove that general complete bipartite graphs are (2, 2)-choosable by this method in [13].…”

Section: Algebraic Methodsmentioning

confidence: 99%

“…It was shown in [7] that complete bipartite graphs other than K 2 are 3-edge weight choosable. By assigning every vertex weight 0, we have the following corollary.…”

Section: Assume the Claim Is Not True Thenmentioning

confidence: 99%

“…The method used in [7] and in [13] for the study of edge weight choosability or total weight choosability of graphs is algebraic (see Section 5), which proves the existence of a proper edge weighting or proper total weighting without actually constructing such a weighting.…”

Section: Introductionmentioning

confidence: 99%

“…Bartnicki, Grytczuk and Niwczykthe [7] considered the choosability version of edge weighting. A graph is said to be k-edge weight choosable if the following is true: For any list assignment L which assigns to each edge e a set L(e) of k real numbers, G has a proper edge weighting f such that f (e) ∈ L(e) for each edge e. They conjectured that every graph without isolated edges is 3-edge weight choosable, and verified the conjecture for complete graphs, complete bipartite graphs and some other graphs.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Section: Algebraic Methodsmentioning

confidence: 99%

“…It was shown in [7] that complete bipartite graphs other than K 2 are 3-edge weight choosable. By assigning every vertex weight 0, we have the following corollary.…”

Section: Assume the Claim Is Not True Thenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

List Total Weighting of Graphs

Wong

Zhu

Yang

2010

Fete of Combinatorics and Computer Science

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“…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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Total weight choosability of graphs

Wong

Zhu

2010

J. Graph Theory

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Abstract:A graph G = (V , E) is called (k, k )-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k real numbers, there is a mapping f : V ∪E → R such that f (y) ∈ L(y) for any y ∈ V ∪E and for any two adjacent vertices x, x , e∈E(x) f (e)+f (x) = e∈E(x ) f (e)+f (x ). We conjecture that every graph is (2, 2)-total weight choosable and every graph without isolated edges is (1, 3)-total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K 2 are (1, 3)-total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)-total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)-total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)-total weight

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Weight choosability of graphs

Cited by 74 publications

References 12 publications

List Total Weighting of Graphs

List Total Weighting of Graphs

Graphs with multiplicative vertex-coloring 2-edge-weightings

Total weight choosability of graphs

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