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Journal of Global Optimization

, Volume 60, Issue 4, pp 777–791 | Cite as

Minimum total coloring of planar graph

  • Huijuan Wang
  • Lidong Wu
  • Weili WuEmail author
  • Panos M. Pardalos
  • Jianliang Wu
Article

Abstract

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is familiar to us by the name of “total coloring”. Total coloring is a coloring of \(V\cup {E}\) such that no two adjacent or incident elements receive the same color. In other words, total chromatic number of \(G\) is the minimum number of disjoint vertex independent sets covering a total graph of \(G\). Here, let \(G\) be a planar graph with \(\varDelta \ge 8\). We proved that if for every vertex \(v\in V\), there exists two integers \(i_{v},j_{v} \in \{3,4,5,6,7,8\}\) such that \(v\) is not incident with intersecting \(i_v\)-cycles and \(j_v\)-cycles, then the vertex chromatic number of total graph of \(G\) is \(\varDelta +1\), i.e., the total chromatic number of \(G\) is \(\varDelta +1\).

Keywords

Planar graph Total coloring Cycle Independent set 

Notes

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 11201440, 11271006, 11301410, the Natural Science Basic Research Plan in Shaanxi Province of China under Grant 2013JQ1002, and the Scientific Research Foundation for the Excellent Young and Middle-Aged Scientists of Shandong Province of China under Grant BS2013DX002. This work was also supported in part by the National Science Foundation of USA under Grants CNS-0831579 and CCF-0728851.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Huijuan Wang
    • 1
  • Lidong Wu
    • 2
  • Weili Wu
    • 2
    • 3
    Email author
  • Panos M. Pardalos
    • 4
  • Jianliang Wu
    • 1
  1. 1.School of MathematicsShandong UniversityJinan China
  2. 2.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA
  3. 3.College of Computer Science and TechnologyTaiYuan University of TechnologyTaiyuan China
  4. 4.Department of Industrial and System Engineering and Center for Applied OptimizationUniversity of FloridaGainesvilleUSA

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