Abstract
Let mad (G) denote the maximum average degree (over all subgraphs) of G and let χ i (G) denote the injective chromatic number of G. We prove that if Δ≥4 and \(\mathrm{mad}(G)<\frac{14}{5}\), then χ i (G)≤Δ+2. When Δ=3, we show that \(\mathrm{mad}(G)<\frac{36}{13}\) implies χ i (G)≤5. In contrast, we give a graph G with Δ=3, \(\mathrm{mad}(G)=\frac{36}{13}\), and χ i (G)=6.
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Research of S.-J. Kim supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00115).
Research of G. Yu supported in part by the NSF grant DMS-0852452.
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Cranston, D.W., Kim, SJ. & Yu, G. Injective Colorings of Graphs with Low Average Degree. Algorithmica 60, 553–568 (2011). https://doi.org/10.1007/s00453-010-9425-x
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DOI: https://doi.org/10.1007/s00453-010-9425-x