Strong edge-coloring for jellyfish graphs

Chang, Gerard J.; Chen, Shenghua; Hsu, Chi-Yun; Hung, Chia-Man; Lai, Huei-ling

doi:10.1016/j.disc.2015.04.031

Cited by 5 publications

(7 citation statements)

References 20 publications

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“…Then H is called a C k -jellyfish. On such graphs, Chang et al [11] proved a general lower bound (their Theorem 13), which implies the following formula.…”

Section: Theorem 6 the Strong Chromatic Index Of Thcmentioning

confidence: 95%

“…Theorem 12 [11] If G is a C k -jellyfish of m edges, such that k is odd and all vertices of C k have the same degree in G, then v 0 s ðGÞ ! d m bk=2c e.…”

Section: Theorem 6 the Strong Chromatic Index Of Thcmentioning

confidence: 99%

“…The exact values for the puffer graphs are obtained in [11]. A Halin graph is a plane graph constructed from a tree T without vertices of degree two by connecting all leaves through a cycle C. Let G ¼ T [ C be a Halin graph.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Then H is called a C k -jellyfish. On such graphs, Chang et al [11] proved a general lower bound (their Theorem 13), which implies the following formula.…”

Section: Theorem 6 the Strong Chromatic Index Of Thcmentioning

confidence: 95%

“…Theorem 12 [11] If G is a C k -jellyfish of m edges, such that k is odd and all vertices of C k have the same degree in G, then v 0 s ðGÞ ! d m bk=2c e.…”

Section: Theorem 6 the Strong Chromatic Index Of Thcmentioning

confidence: 99%

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Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Dara

Mishra

Narayanan

et al. 2022

Graphs and Combinatorics

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show abstract

“…A C n -jellyfish is a graph by adding pendant edges at the vertices of C n . In [9], it is shown that Proposition 24. If G is a C n -jellyfish of m edges with σ(G) ≥ 4, then χ s (G) = Adopting these results leads to a strengthening of Theorem 10.…”

Section: Consequences Concerning the Maximum Average Degreementioning

confidence: 99%

On the precise value of the strong chromatic index of a planar graph with a large girth

Chang

Duh

2018

Discrete Applied Mathematics

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A strong k-edge-coloring of a graph G is a mapping from E(G) to {1, 2, . . . , k} such that every pair of distinct edges at distance at most two receive different colors. The strong chromatic index χ s (G) of a graph G is the minimum k for which G has a strong k-edge-coloring. Denote σ(G) = max xy∈E(G) {deg(x) + deg(y) − 1}. It is easy to see that σ(G) ≤ χ s (G) for any graph G, and the equality holds when G is a tree. For a planar graph G of maximum degree ∆, it was proved that χ s (G) ≤ 4∆ + 4 by using the Four Color Theorem. The upper bound was then reduced to 4∆, 3∆ + 5, 3∆ + 1, 3∆, 2∆ − 1 under different conditions for ∆ and the girth. In this paper, we prove that if the girth of a planar graph G is large enough and σ(G) ≥ ∆(G) + 2, then the strong chromatic index of G is precisely σ(G). This result reflects the intuition that a planar graph with a large girth locally looks like a tree.A Vizing-type problem was asked by Erdős and Nešetřil, and further strengthened by Faudree, Schelp, Gyárfás and Tuza to give an upper bound for χ s (G) in terms of the maximum degree ∆ = ∆(G):Conjecture 1 (Erdős and Nešetřil '88 [16] '89 [17], Faudree et al '90 [18]). If G is a graph with maximum degree ∆, then χ s (G) ≤ ∆ 2 + ∆ 2 2 .As demonstrated in [18], there are indeed some graphs reach the given upper bounds.By a greedy algorithm, it can be easily seen that χ s (G) ≤ 2∆(∆ − 1) + 1. Molloy and Reed [28] using the probabilistic method to show that χ s (G) ≤ 1.998∆ 2 for maximum degree ∆ large enough. Recently, this upper bound was improved by Bruhn and Joos [8] to 1.93∆ 2 .For small maximum degrees, the cases ∆ = 3 and 4 were studied. Andersen [1] and Horák et al [22] proved that χ s (G) ≤ 10 for ∆(G) ≤ 3 independently; and Cranston [13] showed that χ s (G) ≤ 22 when ∆(G) ≤ 4.According to the examples in [18], the bound is tight for ∆ = 3, and the best we may expect for ∆ = 4 is 20.The strong chromatic index of a few families of graphs are examined, such as

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“…Then H is called a C kjellyfish. On such graphs, Chang et al [11] proved a general lower bound (their Theorem 13), which implies the following formula. Theorem 4.8 [11] If G is a C k -jellyfish of m edges, such that k is odd and all vertices of C k have the same degree in G, then χ ′ s (G) ≥ ⌈ m ⌊k/2⌋ ⌉.…”

mentioning

confidence: 95%

Strong edge coloring of Cayley graphs and some product graphs

Dara¹,

Mishra²,

Narayanan³

et al. 2021

Preprint

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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 5 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by 4.

show abstract

Strong edge-coloring for jellyfish graphs

Cited by 5 publications

References 20 publications

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

On the precise value of the strong chromatic index of a planar graph with a large girth

Strong edge coloring of Cayley graphs and some product graphs

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