Strong edge-coloring of planar graphs

Hudak, David E.; Lužar, Borut; Soták, Roman; Škrekovski, Riste

doi:10.1016/j.disc.2014.02.002

Cited by 26 publications

(17 citation statements)

References 10 publications

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“…In this section, we will show several properties of G, including that G is simple, has no small cycles and the distance between any two 2-vertices is at least three, a fact that we will strengthen in a later section. Similar statements are proven in [11,12,14] while considering minimal counterexamples with different properties. Lemma 3.…”

Section: Basic Propertiessupporting

confidence: 75%

Strong chromatic index of subcubic planar multigraphs

Kostochka

Ruksasakchai

et al. 2016

European Journal of Combinatorics

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Abstract. The strong chromatic index of a multigraph is the minimum k such that the edge set can be k-colored requiring that each color class induces a matching. We verify a conjecture of Faudree, Gyárfás, Schelp and Tuza, showing that every planar multigraph with maximum degree at most 3 has strong chromatic index at most 9, which is sharp.This paper is to appear in European J. Combin. 51 (2016) 380-397.Mathematics Subject Classification: 05C15 (05C10)

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Section: Basic Propertiessupporting

confidence: 75%

Strong chromatic index of subcubic planar multigraphs

Kostochka

Ruksasakchai

et al. 2016

European Journal of Combinatorics

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“…This result is best possible since the prism, shown in Figure 1, is a subcubic planar graph with χ ′ s (G) = 9. Several papers prove sharper bounds on the strong chromatic index of planar graphs with additional structure [11,12,13,14], generally by introducing conditions on maximum average degree or girth to ensure that the target graph is sufficiently sparse. For a graph G, the maximum average degree of G, denoted mad(G), is the maximum of average degrees over all subgraphs of G. Hocquard, Montassier, Raspaud, and Valicov [12,13] proved the following.…”

Section: Introductionmentioning

confidence: 99%

On the Strong Chromatic Index of Sparse Graphs

DeOrsey

Diemunsch

Ferrara

et al. 2018

Electron. J. Combin.

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The strong chromatic index of a graph G, denoted χ ′ s (G), is the least number of colors needed to edge-color G so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted χ ′ ℓ,s (G), is the least integer k such that if arbitrary lists of size k are assigned to each edge then G can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if G is a subcubic planar graph with girth(G) ≥ 41 then χ ′ ℓ,s (G) ≤ 5, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759-770]. We further show that if G is a subcubic planar graph and girth(G) ≥ 30, then χ ′ s (G) ≤ 5, improving a bound from the same paper. Finally, if G is a planar graph with maximum degree at most four and girth(G) ≥ 28, then χ ′ s (G) ≤ 7, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.

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“…This study was directly inspired by a theorem of Mahdian , which says that the strong chromatic index of graphs without four‐cycles is of order at most

\frac{Δ^{2}}{\ln Δ}

.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth to mention that Theorem is asymptotically tight; there are graphs without four‐cycles that have strong chromatic index at least

(\frac{1}{2} - ϵ) \frac{Δ^{2}}{\ln Δ}

(see ).…”

Section: Introductionmentioning

confidence: 99%

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Strong chromatic index of unit distance graphs

Dębski

2018

Journal of Graph Theory

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The strong chromatic index of a graph G, denoted by s ′ ( G ), is defined as the least number of colors in a coloring of edges of G, such that each color class is an induced matching (or: if edges e and f have the same color, then both vertices of e are not adjacent to any vertex of f). A graph G is a unit distance graph in R n if vertices of G can be uniquely identified with points in R n, so that u v is an edge of G if and only if the Euclidean distance between the points identified with u and v is 1. We would like to find the largest possible value of s ′ ( G ), where G is a unit distance graph (in R 2 and R 3) of maximum degree Δ. We show that s ′ ( G ) ≤ K Δ 2 ln Δ, where G is a unit distance graph in R 3 of maximum degree Δ. We also show that the maximum possible size of a strong clique in unit distance graph in R 3 is linear in Δ and give a tighter result 6 Δ − 3 for unit distance graphs in the plane.

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Strong edge-coloring of planar graphs

Cited by 26 publications

References 10 publications

Strong chromatic index of subcubic planar multigraphs

Strong chromatic index of subcubic planar multigraphs

On the Strong Chromatic Index of Sparse Graphs

Strong chromatic index of unit distance graphs

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