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Wan, Peng‐Jun

doi:10.1023/a:1009759916586

Cited by 37 publications

(4 citation statements)

References 5 publications

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“…The motivation to study this kind of a coloring came from the hidden terminal problem in telecommunication [10,42]. Optimal (i.e., using the minimum number of colors) L(0, 1)-labelings are known for simple classes of graphs, like paths, cycles, grids (see Makansi [42] and Jin, Yeh [35]), hypercubes (see Wan [60]), and complete binary trees (see Bertossi and Bonuccelli [10]). Bodlaender, Kloks, Tan, van Leeuwen [11] showed some bounds on the number of colors required to find an L(0, 1)-labeling of G for some special classes of graphs: bounded-treewidth graphs, permutation graphs, outerplanar graphs, split graphs, and bipartite graphs.…”

Section: Our Contributionmentioning

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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Section: Our Contributionmentioning

confidence: 99%

Intersecting edge distinguishing colorings of hypergraphs

Okrasa¹,

Rzążewski²

2018

Preprint

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Section: Introductionmentioning

confidence: 99%

“…The kth power of a graph Γ = (V, E) is the graph Γ k = (V ′ , E ′ ), where V ′ = V and in which two vertices are adjacent exactly when their distance in Γ is at most k. In the current work, we focus on the chromatic number of (the cube-like graph) Q k n , denoted by χk(n). The chromatic number χk(n) has been studied, for example, in [9,22,14,16] and is further motivated by the problem of scalability of certain optical networks [21].The value of χ2(n) has been determined for n ≤ 7, and for n = 8 it has been known that 13 ≤ χ2(8) ≤ 14, where the upper bound follows from 14-colorings found independently by Hougardy [22] and Royle [6, Section 9.7]. By finding a 13-coloring of Q 2 8 , we shall here prove that χ2(8) = 13.…”

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confidence: 99%

The chromatic number of the square of the $8$-cube

Kokkala

Östergård

2017

Math. Comp.

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A cube-like graph is a Cayley graph for the elementary abelian group of order 2 n . In studies of the chromatic number of cube-like graphs, the kth power of the n-dimensional hypercube, Q k n , is frequently considered. This coloring problem can be considered in the framework of coding theory, as the graph Q k n can be constructed with one vertex for each binary word of length n and edges between vertices exactly when the Hamming distance between the corresponding words is at most k. Consequently, a proper coloring of Q k n corresponds to a partition of the n-dimensional binary Hamming space into codes with minimum distance at least k + 1. The smallest open case, the chromatic number of Q 2 8 , is here settled by finding a 13-coloring. Such 13-colorings with specific symmetries are further classified.

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“…Then 0,1 n and there exists a labelling scheme using such a number of labels. This labelling is optimal when for some and it is a 2-approximation otherwise [9]. For a bipartite -labelling of cartesian product of a cycle and path is done by Chiang and Yan [11].…”

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confidence: 99%

L(0, 1)-Labelling of Cactus Graphs

Khan¹,

Pal²,

Pal³

2012

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An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G

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Cited by 37 publications

References 5 publications

Intersecting edge distinguishing colorings of hypergraphs

Intersecting edge distinguishing colorings of hypergraphs

The chromatic number of the square of the $8$-cube

<i>L</i>(0, 1)-Labelling of Cactus Graphs

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Untitled

Cited by 37 publications

References 5 publications

Intersecting edge distinguishing colorings of hypergraphs

Intersecting edge distinguishing colorings of hypergraphs

The chromatic number of the square of the $8$-cube

&lt;i&gt;L&lt;/i&gt;(0, 1)-Labelling of Cactus Graphs

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<i>L</i>(0, 1)-Labelling of Cactus Graphs