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Journal of Combinatorial Optimization

, Volume 36, Issue 1, pp 23–34 | Cite as

Neighbor sum distinguishing total coloring of graphs with bounded treewidth

  • Miaomiao Han
  • You Lu
  • Rong Luo
  • Zhengke Miao
Article
  • 75 Downloads

Abstract

A proper total k-coloring \(\phi \) of a graph G is a mapping from \(V(G)\cup E(G)\) to \(\{1,2,\dots , k\}\) such that no adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let \(m_{\phi }(v)\) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if \(m_{\phi }(u)\not =m_{\phi }(v)\) for each edge \(uv\in E(G).\) Let \(\chi _{\Sigma }^t(G)\) be the neighbor sum distinguishing total chromatic number of a graph G. Pilśniak and Woźniak conjectured that for any graph G, \(\chi _{\Sigma }^t(G)\le \Delta (G)+3\). In this paper, we show that if G is a graph with treewidth \(\ell \ge 3\) and \(\Delta (G)\ge 2\ell +3\), then \(\chi _{\Sigma }^t(G)\le \Delta (G)+\ell -1\). This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when \(\ell =3\) and \(\Delta \ge 9\), we show that \(\Delta (G) + 1\le \chi _{\Sigma }^t(G)\le \Delta (G)+2\) and characterize graphs with equalities.

Keywords

Total coloring Neighbor sum distinguishing Treewidth 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematical ScienceTianjin Normal UniversityTianjinPeople’s Republic of China
  2. 2.Department of Applied Mathematics, School of ScienceNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  3. 3.Department of MathematicsWest Virginia UniversityMorgantownUSA
  4. 4.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China

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