# Frontiers of Mathematics in China

 Front. Math. China    2017, Vol. 12 Issue (4) : 937-947     DOI: 10.1007/s11464-017-0649-9
 RESEARCH ARTICLE |
Neighbor sum distinguishing total chromatic number of K4-minor free graph
Hongjie SONG, Changqing XU()
School of Science, Hebei University of Technology, Tianjin 300401, China
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 Abstract A k-total coloring of a graph G is a mapping φ: V (G) ∪ E(G) →{1, 2, . . . , k} such that no two adjacent or incident elements in V (G) ∪ 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 φ is a k-neighbor sum distinguishing total coloring of G if f(u) ≠ f(v) for each edge uv ∈ E(G). Denote $XΣ''(G)$ the smallest value k in such a coloring of G. Pilśniak andWoźniak conjectured that for any simple graph with maximum degree Δ(G), $XΣ''(G)≤Δ(G)+3$. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K4-minor free graph G with Δ(G)≥5, $XΣ''(G)=Δ(G)+1$ if G contains no two adjacent Δ-vertices, otherwise, $XΣ''(G)=Δ(G)+2$. Corresponding Authors: Changqing XU Issue Date: 06 July 2017
 Cite this article: Hongjie SONG,Changqing XU. Neighbor sum distinguishing total chromatic number of K4-minor free graph[J]. Front. Math. China, 2017, 12(4): 937-947. URL: http://journal.hep.com.cn/fmc/EN/10.1007/s11464-017-0649-9 http://journal.hep.com.cn/fmc/EN/Y2017/V12/I4/937