Abstract
A k-edge coloring \(\varphi \) of a graph G is injective if \(\varphi (e_1)\ne \varphi (e_3)\) for any three consecutive edges \(e_1, e_2\) and \(e_3\) of a path or a triangle. The injective chromatic index \(\chi _i'(G)\) of G is the smallest k such that G admits an injective k-edge coloring. By discharging method, we demonstrate that any graph with maximum degree \(\Delta \le 5\) has \(\chi _i'(G)\le 12\) (resp. 13) if its maximum average degree is less than \(\frac{20}{7}\) (resp. 3), which improves the results of Zhu (2023).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
No data was used for the research described in the article.
References
Borodin OV, Kostochka AV, Nešetřil J, Raspaud A, Sopena E (1999) On the maximum average degree and the oriented chromatic number of a graph. Discrete Math 206:77–89
Bu YH, Qi CT (2018) Injective edge coloring of sparse graphs. Discrete Math Algorithms Appl 10(2):1850022
Cardoso DM, Cerdeira JO, Cruz JP, Dominic C (2019) Injective edge coloring of graphs. Filomat 33(19):6411–6423
Ferdjallah B, Kerdjoudj S, Raspaud A (2020) Injective edge-coloring of sparse graphs. arXiv:1907.09838v2 [math.CO]
Foucaud F, Hocquard H, Lajou D (2021) Complexity and algorithms for injective edge-coloring in graphs. Inf Process Lett 170:106121
Hu XL, Legass BM (2023) Injective edge chromatic index of generalized Petersen graphs. Bull Malays Math Sci Soc 46:37
Kostochka A, Raspaud A, Xu JW (2021) Injective edge-coloring of graphs with given maximum degree. European J Combin 96:103355
Lu J, Liu HQ, Hu XL (2022) Injective edge coloring for graphs with small edge weight. Graphs Combin 38:160
Miao ZK, Song YM, Yu G (2022) Note on injective edge-coloring of graphs. Discrete Appl Math 310:65–74
Yue J, Zhang SL, Zhang X (2016) Note on the perfect EIC-graphs. Appl Math Comput 289:481–485
Zhu JL (2023) Injective edge coloring of graphs with maximum degree 5. Discrete Appl Math 334:119–126
Acknowledgements
This work is supported by National Natural Science Foundation of China (No. 12371338), Anhui Provincial Natural Science Foundation (No. 2108085MA01), Outstanding Youth Scientific Research Projects of Anhui Provincial Department of Education (No. 2022AH030073) and Key Projects in Natural Science Research of Anhui Provincial Department of Education (No. 2023AH050268, 2022AH050594).
Funding
National Natural Science Foundation of China (No. 12371338).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lu, J., Hong, ZM. & Xia, ZJ. On injective chromatic index of sparse graphs with maximum degree 5. J Comb Optim 48, 41 (2024). https://doi.org/10.1007/s10878-024-01234-7
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-024-01234-7