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Color Degree Sum Conditions for Rainbow Triangles in Edge-Colored Graphs

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Abstract

Let G be an edge-colored graph and v a vertex of G. The color degree of v is the number of colors appearing on the edges incident to v. A rainbow triangle in G is one in which all edges have distinct colors. In this paper, we first prove that an edge-colored graph on n vertices contains a rainbow triangle if the color degree sum of every two adjacent vertices is at least \(n+1\). Afterwards, we characterize the edge-colored graphs on n vertices containing no rainbow triangles but satisfying that each pair of adjacent vertices has color degree sum at least n.

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Acknowledgments

The authors are grateful to the referees for their helpful comments.

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Ruonan Li & Shenggui Zhang

  2. Faculty of EEMCS, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Ruonan Li

  3. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Bo Ning

Authors
  1. Ruonan Li
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  2. Bo Ning
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  3. Shenggui Zhang
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Corresponding author

Correspondence to Shenggui Zhang.

Additional information

Supported by NSFC (No. 11271300) and China Scholarship Council (No. 201506290097).

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Li, R., Ning, B. & Zhang, S. Color Degree Sum Conditions for Rainbow Triangles in Edge-Colored Graphs. Graphs and Combinatorics 32, 2001–2008 (2016). https://doi.org/10.1007/s00373-016-1690-2

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  • DOI: https://doi.org/10.1007/s00373-016-1690-2

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