RESEARCH ARTICLE

Neighbor sum distinguishing total colorings of K4-minor free graphs

  • Hualong LI ,
  • Bingqiang LIU ,
  • Guanghui WANG
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  • School of Mathematics, Shandong University, Jinan 250100, China

Received date: 15 Jun 2012

Accepted date: 20 Jun 2013

Published date: 01 Dec 2013

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

A total [k]-coloring of a graph G is a mapping ϕ: V (G) ∪E(G) → {1,2, ..., k} such that any two adjacent elements in V (G)∪E(G) receive different colors. Let f(v) denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uvE(G),f(u)f(v). By χnsd(G), we denote the smallest value k in such a coloring of G. Pilśniak and Woźniak conjectured χnsd(G)(G)+3 for any simple graph with maximum degree Δ(G). This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for K4-minor free graphs. Furthermore, we show that if G is a K4-minor free graph with (G)4, then χnsd(G)(G)+2 The bound Δ(G) + 2 is sharp.

Cite this article

Hualong LI , Bingqiang LIU , Guanghui WANG . Neighbor sum distinguishing total colorings of K4-minor free graphs[J]. Frontiers of Mathematics in China, 0 , 8(6) : 1351 -1366 . DOI: 10.1007/s11464-013-0322-x

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