On strong edge-coloring of graphs with maximum degree 4

Lv, Jian-Bo; Li, Xiangwen; Yu, Gexin

doi:10.1016/j.dam.2017.09.006

Cited by 11 publications

(4 citation statements)

References 10 publications

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“…In 2015, Bensmail, Bonamy and Hocquard [2] studied the upper bound on χ s (G) in terms of maximum average degree for the class of graphs with maximum degree 4, where they gave some sufficient conditions under which Conjecture 1.1 is true. In 2018, Lv, Li and Yu [17] strengthened Bensmail, Bonamy and Hocquard's results. And they proved that for any graph G with ∆(G) = 4, if there are two vertices of degree 3 whose distance is at most 4, then χ s (G) ≤ 20.…”

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confidence: 79%

The tight bound for the strong chromatic indices of claw-free subcubic graphs

Lin¹,

Lin²

2022

Preprint

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Let G be a graph and k a positive integer. A strong k-edge-coloring of G is a mapping φ : E(G) → {1, 2, . . . , k} such that for any two edges e and e that are either adjacent to each other or adjacent to a common edge, φ(e) = φ(e ). The strong chromatic index of G, denoted as χ s (G), is the minimum integer k such that G has a strong k-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38 (3) ( 2022) 63] proved that if G is a claw-free subcubic graph other than the triangular prism then χ s (G) ≤ 8. In addition, they asked if the upper bound 8 can be improved to 7. In this paper, we answer this question in the affirmative. Our proof implies a linear-time algorithm for finding strong 7-edge-colorings of such graphs. We also construct infinitely many claw-free subcubic graphs with their strong chromatic indices attaining the bound 7.

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confidence: 79%

The tight bound for the strong chromatic indices of claw-free subcubic graphs

Lin¹,

Lin²

2022

Preprint

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“…This was followed by a significant contribution published by Erdős and Nešetřil [2], who formally defined the problem. Over the past years, analysis of the parameter χ s (G) has focused on various attributes of graph G. Contributions by Lv et al [3] and Huang et al [4] determined the bounds for χ s (G) based on the maximum degree ∆ G of the graph G. In particular, the article by Huang et al established an upper bound for χ s (G) for every planar graph with a maximum degree 4. The article by Lv et al obtained results for the upper bound of χ s (G) based on a new attribute called the maximum average degree of graph G. This article established the proof using a partition of the vertex set of graph G. The influence of the planar nature of graphs on χ s (G) was explored in the article [5] by Chang and Duh, and in the article [6] by Wang et al The contributions made by Chang and Duh classified graphs based on their girth and the maximum degree ∆ of the graph.…”

Section: Factors Influencing χ S (G)mentioning

confidence: 99%

An Analysis of the Factors Influencing the Strong Chromatic Index of Graphs Derived by Inflating a Few Common Classes of Graphs

Vikram

Balaji

2023

Symmetry

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The problem of strong edge coloring discusses assigning colors to the edges of a graph such that distinct colors are assigned to any two edges which are either adjacent to each other or are adjacent to a common edge. The least number of colors required to define a strong edge coloring of a graph is called its strong chromatic index. This problem is equivalent to the problem of assigning collision-free frequencies to the links between the elements of a wireless sensor network. In this article, we discuss a novel way of generating new graphs from existing graphs. This graph construction is known as inflating a graph. We discuss the strong chromatic index of graphs generated by inflating some common classes of graphs and graphs derived from them. In particular, we consider the cycle graph, which is symmetric in nature, and graphs such as the path graph and the star graph, which are not symmetric. Further, we analyze the factors which influence the strong chromatic index of these inflated graphs.

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“…(The case of non-regular sub-cubic graphs is easy, as noted in [17]). The case of D ¼ 4 is already complicated, the upper bound of 20 (respectively 19,18,17,16) is known to be valid only under the further assumption that the maximum average degree is at most 51/13 (respectively 15/4, 18/5, 7/2, 61/18) [30]. These are improvements of the estimates in [4], where e.g.…”

Section: Introductionmentioning

confidence: 99%

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Dara

Mishra

Narayanan

et al. 2022

Graphs and Combinatorics

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A strong edge coloring of a graph G is a proper edge coloring of G such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 4 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by 4.

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On strong edge-coloring of graphs with maximum degree 4

Cited by 11 publications

References 10 publications

The tight bound for the strong chromatic indices of claw-free subcubic graphs

The tight bound for the strong chromatic indices of claw-free subcubic graphs

An Analysis of the Factors Influencing the Strong Chromatic Index of Graphs Derived by Inflating a Few Common Classes of Graphs

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

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