A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2,...,k}. Let f（u） denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neigh...A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2,...,k}. Let f（u） denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E（G）, f（u） ≠ f（v）. By X＂nsd（G）, we denote the smallest value k in such a coloring of G. Pilgniak and Wozniak conjectured that X＂nsd（G） ≤ △（G）＋ 3 for any simple graph with maximum degree △（G）. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11201180)the Scientific Research Foundation of University of Ji’nan(Grant No.XKY1120)
文摘A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2,...,k}. Let f（u） denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E（G）, f（u） ≠ f（v）. By X＂nsd（G）, we denote the smallest value k in such a coloring of G. Pilgniak and Wozniak conjectured that X＂nsd（G） ≤ △（G）＋ 3 for any simple graph with maximum degree △（G）. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.
