Abstract
A coloring of graph G is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number χi(G) of G is the least integer k such that G has an injective k-coloring. In this paper, we prove that (1) if G is a planar graph with girth g ≥ 6 and maximum degree Δ ≥ 7, then χi(G) ≤ Δ + 2; (2) if G is a planar graph with Δ ≥ 24 and without 3,4,7-cycles, then χi(G) ≤ Δ + 2.
This is a preview of subscription content, log in via an institution to check access.
References
Borodin, O.V., Ivanova, A.O. Injective (Δ + 1)-coloring of planar graph with girth 6. Siberian Mathematical Journal, 52: 23–29 (2011)
Doyon, A., Hahn, G., Raspaud, A. On the injective chromatic number of sparse graphs. preprint, (2005)
Doyon, A., Hahn, G., Raspaud, A. Some bounds on the injective chromatic number of graphs. Discrete Mathematics, 310: 585–590 (2005)
Bu, Y., Lu, K. List injective coloring of planar graphs with girth 5,6,8. Discrete Applied Mathematics, 161: 1367–1377 (2013)
Dong, W., Lin, W. Injective coloring of planar graph with girth 6. Discrete Mathematics, 313: 1302–1311 (2013)
Cranston, D.W., Kim, S.-J., Yu, G. Injective colorings of sparse graphs. Discrete Mathematics, 310: 2965–2973 (2010)
Bu, Y., Lu, K. Injective coloring of planar graphs with girth 7. Discrete Mathematics, Algorithms and Applications, 4: 1250034 (2012)
Bu, Y., Lu, K. Yang, S. Two smaller upper bounds of list injective chromatic number. Journal of Combinatorial Optimization, 29(2): 373–388 (2015)
Bu, Y., Wang, C., Yang, S. List injective Coloring of Planar Graphs. ARS COMBINATORIA Discrete Mathematics, Algorithms and Applications, 141: 191–211 (2018)
Dong, W., Lin, W. Injective coloring of plane graphs with girth 5. Discrete Mathematics, 315: 120–127 (2014)
Acknowledgments. The authors appreciate the anonymous reviewers for their careful comments for this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflict of interest.
Additional information
This paper is supported by the National Natural Science Foundation of China (No. 11871377).
Rights and permissions
About this article
Cite this article
Chen, Y., Tao, L. & Zhang, L. Injective Δ+2 Coloring of Planar Graph Without Short Cycles. Acta Math. Appl. Sin. Engl. Ser. 39, 1009–1031 (2023). https://doi.org/10.1007/s10255-023-1098-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-023-1098-8