摘要
A k-total coloring of a graph G is a mapping φ： V（G） U E（G） → {1, 2,..., k} such that no two adjacent or incident elements in V（G） U E（G） receive the same color. Let f（v） denote the sum of the color on the vertex v and the colors on all edges incident with v. We say that ~ is a k-neighbor sum distinguishing total coloring of G if f（u） ≠ f（v） for each edge uv C E（G）. Denote X＂ （G） the smallest value k in such a coloring of G. Pilgniak and Wo/niak conjectured that for any simple graph with maximum degree △（G）, X＂（G） ≤ 3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for Ka-minor free graph G with △（G） ≥ 5, X＂（G） = △（G） ＋ 1 if G contains no two adjacent A-vertices, otherwise, X＂（G） = △（G） ＋ 2.
A k-total coloring of a graph G is a mapping φ： V（G） U E（G） → {1, 2,..., k} such that no two adjacent or incident elements in V（G） U E（G） receive the same color. Let f（v） denote the sum of the color on the vertex v and the colors on all edges incident with v. We say that ~ is a k-neighbor sum distinguishing total coloring of G if f（u） ≠ f（v） for each edge uv C E（G）. Denote X＂ （G） the smallest value k in such a coloring of G. Pilgniak and Wo/niak conjectured that for any simple graph with maximum degree △（G）, X＂（G） ≤ 3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for Ka-minor free graph G with △（G） ≥ 5, X＂（G） = △（G） ＋ 1 if G contains no two adjacent A-vertices, otherwise, X＂（G） = △（G） ＋ 2.
