The Tight Bound for the Strong Chromatic Indices of Claw-Free Subcubic Graphs

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 ′ (...

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Published in:	Graphs and combinatorics Vol. 39; no. 3; p. 58
Main Authors:	Lin, Yuquan, Lin, Wensong
Format:	Journal Article
Language:	English
Published:	Tokyo Springer Japan 01.05.2023 Springer Nature B.V
Subjects:	Algorithms Combinatorial analysis Combinatorics Engineering Design Graph coloring Graphs Integers Mathematics Mathematics and Statistics Original Paper Polynomials Upper bounds Subcubic graph Strong chromatic index Strong edge coloring Claw-free graph
ISSN:	0911-0119, 1435-5914
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Abstract	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 polynomial-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.
AbstractList	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 χs′(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 χs′(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 polynomial-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. 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 polynomial-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.
ArticleNumber	58
Author	Lin, Wensong Lin, Yuquan
Author_xml	– sequence: 1 givenname: Yuquan surname: Lin fullname: Lin, Yuquan organization: School of Mathematics, Southeast University – sequence: 2 givenname: Wensong orcidid: 0000-0002-4112-0469 surname: Lin fullname: Lin, Wensong email: wslin@seu.edu.cn organization: School of Mathematics, Southeast University
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Cites_doi	10.1002/jgt.3190170204 10.1109/INFCOM.1997.644573 10.1006/jctb.1997.1724 10.1016/0012-365X(92)90678-9 10.1016/j.endm.2015.06.104 10.1016/j.dam.2017.09.006 10.1016/0012-365X(88)90196-3 10.1016/j.disc.2018.02.025 10.1016/j.jctb.2022.01.009 10.1016/j.ejc.2015.07.002 10.1112/jlms/s1-10.37.26 10.1016/j.dam.2020.03.024 10.1016/j.endm.2015.06.038 10.1007/s00373-022-02462-6 10.1016/j.disc.2006.03.053 10.1016/j.disc.2007.11.051 10.1137/1.9781611976465.10 10.37236/7016 10.1145/345910.345925
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Snippet	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 ′... 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...
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SubjectTerms	Algorithms Combinatorial analysis Combinatorics Engineering Design Graph coloring Graphs Integers Mathematics Mathematics and Statistics Original Paper Polynomials Upper bounds
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Title	The Tight Bound for the Strong Chromatic Indices of Claw-Free Subcubic Graphs
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