Directed Graphs

  • Xueliang Li
  • Colton Magnant
  • Zhongmei Qin
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


Much like the undirected version, a strongly connected directed graph is called proper connected if between every ordered pair of vertices, there is a directed properly colored path. Defined in Magnant et al. (Matematiqki Vesnik 68(1):58–65, 2016), the directed proper connection number of a strongly connected directed graph G, denoted by \(\overrightarrow {pc}(G)\), is the minimum number of colors needed to color the (directed) edges so that the directed graph is proper connected. Clearly \(\overrightarrow {pc}(G) \geq 2\) for any G since a directed edge from u to v implies there is no directed edge from v to u so a directed path from v to u must use at least 2 colors.


  1. 29.
    Ducoffe, G., Marinescu-Ghemeci, R., Popa, A.: On the (di)graphs with (directed) proper connection number two. Electron. Notes Discrete Math. 62, 237–242 (2017)MathSciNetCrossRefGoogle Scholar
  2. 59.
    Magnant, C., Morley, P.R., Porter, S., Salehi Nowbandegani, P., Wang, H.: Directed proper connection of graphs. Matematiqki Vesnik 68(1), 58–65 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Xueliang Li
    • 1
  • Colton Magnant
    • 2
  • Zhongmei Qin
    • 3
  1. 1.Center for CombinatoricsNankai UniversityTianjinChina
  2. 2.Department of MathematicsGeorgia Southern UniversityStatesboroUSA
  3. 3.College of ScienceChang’an UniversityXi’anChina

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