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 e...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 from the sum of colors taken on the edges incident to v. Let X（G ） denote the smallest value k in such a ＇ G 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 then x＇（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.展开更多
基金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)the Fundamental Research Funds of Shandong University and Independent Innovation Foundation of Shandong University(Grant No.IFYT14012)
文摘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 from the sum of colors taken on the edges incident to v. Let X（G ） denote the smallest value k in such a ＇ G 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 then x＇（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.
