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 \(e_3\) in the same path or triangle. The injective chromatic index \(\chi _i'(G)\) is the smallest k necessary for an injective k-edge coloring of G. Let \({\text {mad}}(G)=\max \{\frac{2|E(H)|}{|V(H)|}:H\subseteq G\}\). We prove that every subcubic graph G has \(\chi _i'(G)\le 6\) if \({\text {mad}}(G)<\frac{30}{11}\), which improves the result of Ferdjallah et al. (Injective edge-coloring of sparse graphs, 2020). We also prove that every graph G with maximum degree 4 has \(\chi _i'(G)\le 12\) if \({\text {mad}}(G)<\frac{33}{10}\), which improves the result of Miao et al. (Discrete Appl Math 310:65–74, 2022).
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References
Bu, Y.H., Qi, C.T.: Injective edge coloring of sparse graphs. Discrete Math. Algorithms Appl. 10(2), 1850022 (2018)
Cardoso, D.M., Cerdeira, J.O., Cruz, J.P., Dominic, C.: Injective edge coloring of graphs. Filomat 33, 6411–6423 (2019)
Ferdjallah, B., Kerdjoudj, S., Raspaud, A.: Injective edge-coloring of sparse graphs. arXiv:1907.09838v2 [math.CO] (2020)
Foucaud, F., Hocquard, H., Lajou, D.: Complexity and algorithms for injective edge-coloring in graphs. Inf. Process. Lett. 170, 106121 (2021)
Hu, X.L., Legass, B.M.: Injective edge chromatic index of generalized Petersen graphs. Bull. Malays. Math. Sci. Soc. 46, 37 (2022)
Kostochka, A., Raspaud, A., Xu, J.W.: Injective edge-coloring of graphs with given maximum degree. Eur. J. Combin. 96, 103355 (2021)
Li, Y., Chen, L.: Injective edge coloring of generalized Petersen graphs. AIMS Math. 6(8), 7929–7943 (2021)
Lu, J., Liu, H.Q., Hu, X.L.: Injective edge coloring for graphs with small edge weight. Graphs Combin. 38, 160 (2022)
Miao, Z.K., Song, Y.M., Yu, G.: Note on injective edge-coloring of graphs. Discrete Appl. Math. 310, 65–74 (2022)
Yue, J., Zhang, S.L., Zhang, X.: Note on the perfect EIC-graphs. Appl. Math. Comput. 289, 481–485 (2016)
Zhu, J.L., Bu, Y.H., Zhu, H.G.: Injective edge coloring of sparse graphs with maximum degree 5. J. Comb. Optim. 45, 46 (2022)
Acknowledgements
This work is supported by Anhui Provincial Natural Science Foundation (No. 2108085MA01, 2108085MA02) and Outstanding Youth Scientific Research Projects of Anhui Provincial Department of Education (No. 2022AH030073).
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Lu, J., Pan, XF. Injective edge coloring of some sparse graphs. J. Appl. Math. Comput. 69, 3421–3431 (2023). https://doi.org/10.1007/s12190-023-01888-2
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DOI: https://doi.org/10.1007/s12190-023-01888-2