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Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree

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Abstract

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] = {1, 2, …, h}. Let w(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge u v ∈ E(G), if w(u) ≠ w(v), then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G. By tndiΣ (G), we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad(G) < 3, then tndiΣ (G) ≤ k + 2 where k = max{Δ(G), 5}. It partially confirms the conjecture proposed by Pilśniak and Woźniak.

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Authors and Affiliations

  1. School of Science, Shandong Jiao Tong University, Ji’nan, 250023, P. R. China

    Ai Jun Dong

  2. School of Mathematics, Shandong University, Ji’nan, 250100, P. R. China

    Guang Hui Wang

Authors
  1. Ai Jun Dong
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  2. Guang Hui Wang
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Correspondence to Guang Hui Wang.

Additional information

The first author is supported by National Natural Science Foundation of China (Grant No. 11161035) and the Research Fund for the Doctoral Program of Shandong Jiaotong University; the second author is supported by National Natural Science Foundation of China (Grant No. 11101243), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100131120017) and the Scientific Research Foundation for the Excellent Middle-Aged and Youth Scientists of Shandong Province of China (Grant No. BS2012SF016)

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Dong, A.J., Wang, G.H. Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree. Acta. Math. Sin.-English Ser. 30, 703–709 (2014). https://doi.org/10.1007/s10114-014-2454-7

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  • DOI: https://doi.org/10.1007/s10114-014-2454-7

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