Abstract
A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different fromthe sumof colors taken on the edges incident to v. Let χ′Σ(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) < 5/2, then χ′Σ(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.
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Supported by National Natural Science Foundation of China (Grant Nos. 11371355, 11471193, 11271006, 11631014), the Foundation for Distinguished Young Scholars of Shandong Province (Grant No. JQ201501) and the Fundamental Research Funds of Shandong University and Independent Innovation Foundation of Shandong University (Grant No. IFYT14012)
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Yu, X.W., Wang, G.H., Wu, J.L. et al. Neighbor sum distinguishing edge coloring of subcubic graphs. Acta. Math. Sin.-English Ser. 33, 252–262 (2017). https://doi.org/10.1007/s10114-017-5516-9
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DOI: https://doi.org/10.1007/s10114-017-5516-9
Keywords
- Proper edge coloring
- neighbor sum distinguishing edge coloring
- maximum average degree
- subcubic graph
- planar graph