On distance edge-colourings and matchings

Kang, Ross J.; Manggala, Putra

doi:10.1016/j.endm.2009.07.049

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…To prove (5), we start by counting all walks in G of length at most t whose first vertex, say u, is in A. By (7), there are at least (2 − 2δ)∆ t−1 choices for u, and so the number of such walks is at least (2 − 2δ)∆ 2t−1 . Of these walks, those whose last vertex is outside of A ∪ B t we call bad.…”

Section: A General Upper Boundmentioning

confidence: 99%

“…Lemma 16 (Kang and Manggala [7]). Let ε > 0 and suppose p = d/n with d ≥ d 0 for some large fixed d 0 .…”

Section: Claim 14 At Most One Endvertex Of a Light Edge Is Inmentioning

confidence: 99%

“…They pointed out a simple class of graphs (blown-up 5-cycles) of arbitrarily large maximum degree ∆ with strong chromatic index 5∆ 2 /4. For t ≥ 3, the second author together with Putra Manggala [7] recently observed that there are examples (in particular, some specific Hamming graphs) that are regular of arbitrarily large degree ∆ and have distance-t chromatic index greater than ∆ t / (2(t − 1) t−1 ) = Ω(∆ t ), thus certifying that the trivial upper bound cannot be replaced by any bound that is o(∆ t ).…”

Section: Introductionmentioning

confidence: 99%

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The Distance-t Chromatic Index of Graphs

Kaiser

Kang

2013

Combinator. Probab. Comp.

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We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2 − ε)∆ t for graphs of maximum degree at most ∆, where ε is some absolute positive constant independent of t. The other is a bound of O(∆ t / log ∆) (as ∆ → ∞) for graphs of maximum degree at most ∆ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree ∆, with distance-t chromatic index at least Ω(∆ t / log ∆).

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Section: A General Upper Boundmentioning

confidence: 99%

“…Lemma 16 (Kang and Manggala [7]). Let ε > 0 and suppose p = d/n with d ≥ d 0 for some large fixed d 0 .…”

Section: Claim 14 At Most One Endvertex Of a Light Edge Is Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Distance-t Chromatic Index of Graphs

Kaiser

Kang

2013

Combinator. Probab. Comp.

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“…The authors also proposed, in the same article, a two‐approximation polynomial‐time algorithm to find an ℓ ‐distance edge coloring of a given planar graph G with at most 2χ ℓ ′( G ) colors. In [22], the authors study the ℓ ‐distance edge coloring for sparse random graphs. In [13], the authors studied the ℓ ‐chromatic index of the Kronecker product of two paths, two cycles, a path by a cycle, two stars, and the graph K 2 by other graphs.…”

Section: Related Workmentioning

confidence: 99%

Distance edge coloring and collision‐free communication in wireless sensor networks

Drira¹,

Effantin²,

Kheddouci³

2013

Networks

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Motivated by the problem of link scheduling in wireless sensor networks where different sensors have different transmission and interference ranges and may be mobile, we study the problem of "distance edge coloring" of graphs, which is a generalization of proper edge coloring. Let G be a graph modeling a sensor network. An -distance edge coloring of G is a coloring of the edges of G such that any two edges within distance of each other are assigned different colors. The parameter is chosen, so that the links corresponding to two edges that are assigned the same color do not interfere. We investigate the -distance edge coloring problem on several families of graphs that can be used as topologies in sensor deployment. We focus on determining the minimum number of colors needed and optimal coloring algorithms.

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“…In particular, there has been considerable interest in χ(L(G) t ) (where χ(H) denotes the chromatic number of H), especially for G of bounded maximum degree. For t = 1, this is the usual chromatic index of G; for t = 2, it is known as the strong chromatic index of G, and is associated with a more famous problem of Erdős and Nešetřil [9]; for t > 2, the parameter is referred to as the distance-t chromatic index, with the study of bounded degree graphs initiated in [13]. We note that the output of Theorem 8 may be directly used as input to a recent result [11] related to Reed's conjecture [15] to bound χ(L(G) t ).…”

Section: Introductionmentioning

confidence: 99%

Maximising line subgraphs of diameter at most $t$

Cambie¹,

Batenburg²,

Verclos³

et al. 2021

Preprint

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We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree-diameter problem. Our main result is that, for any graph of maximum degree ∆ with more than 1.5∆ t edges, its line graph must have diameter larger than t. In the case where the graph contains no cycle of length 2t + 1, we can improve the bound on the number of edges to one that is exact for t ∈ {1, 2, 3, 4, 6}. In the case ∆ = 3 and t = 3, we obtain an exact bound. Our results also have implications for the related problem of bounding the distance-t chromatic index, t > 2; in particular, for this we obtain an upper bound of 1.941∆ t for graphs of large enough maximum degree ∆, markedly improving upon earlier bounds for this parameter.

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On distance edge-colourings and matchings

Cited by 3 publications

References 8 publications

The Distance-t Chromatic Index of Graphs

The Distance-t Chromatic Index of Graphs

Distance edge coloring and collision‐free communication in wireless sensor networks

Maximising line subgraphs of diameter at most $t$

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