Abstract
Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′∑(G). Flandrin et al. proposed the following conjecture that χ′∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < \(\tfrac{{37}} {{12}}\) and Δ(G) ≥ 7.
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This work was supported by the National Natural Science Foundation of China (Grant No. 11571258), the National Natural Science Foundation of Shandong Province (Grant No. ZR2016AM01) and Scientific Research Foundation of University of Jinan (Grant Nos. XKY1414 and XKY1613)
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Qiu, B.J., Wang, J.H. & Liu, Y. Neighbor sum distinguishing colorings of graphs with maximum average degree less than \(\tfrac{{37}} {{12}}\). Acta. Math. Sin.-English Ser. 34, 265–274 (2018). https://doi.org/10.1007/s10114-017-6491-x
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DOI: https://doi.org/10.1007/s10114-017-6491-x
Keywords
- Neighbor sum distinguishing coloring
- combinatorial nullstellensatz
- maximum average degree
- proper colorings