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Strong Edge-Coloring Of Planar Graphs

100%
Song W.-Y., Miao L.-Y.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 4
845-857
EN
A strong edge-coloring of a graph is a proper edge-coloring where each color class induces a matching. We denote by 𝜒's(G) the strong chromatic index of G which is the smallest integer k such that G can be strongly edge-colored with k colors. It is known that every planar graph G has a strong edge-coloring with at most 4 Δ(G) + 4 colors [R.J. Faudree, A. Gyárfás, R.H. Schelp and Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205–211]. In this paper, we show that if G is a planar graph with g ≥ 5, then 𝜒's(G) ≤ 4(G) − 2 when Δ(G) ≥ 6 and 𝜒's(G) ≤ 19 when Δ(G) = 5, where g is the girth of G.
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Upper Bounds for the Strong Chromatic Index of Halin Graphs

100%
Hu Z., Lih K.-W., Liu D. D.-F.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
|
nr 1
5-26
EN
The strong chromatic index of a graph G, denoted by χ′s(G), is the minimum number of vertex induced matchings needed to partition the edge set of G. Let T be a tree without vertices of degree 2 and have at least one vertex of degree greater than 2. We construct a Halin graph G by drawing T on the plane and then drawing a cycle C connecting all its leaves in such a way that C forms the boundary of the unbounded face. We call T the characteristic tree of G. Let G denote a Halin graph with maximum degree Δ and characteristic tree T. We prove that χ′s(G) ⩽ 2Δ + 1 when Δ ⩾ 4. In addition, we show that if Δ = 4 and G is not a wheel, then χ′s(G) ⩽ χ′s(T) + 2. A similar result for Δ = 3 was established by Lih and Liu [21].
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Three edge-coloring conjectures

75%
Schelp R.
Discussiones Mathematicae Graph Theory
|
2002
|
tom 22
|
nr 1
173-182
EN
The focus of this article is on three of the author's open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold.
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