Strongly n-e.c. Graphs and Independent Distinguishing Labellings

  • Christopher DuffyEmail author
  • Jeannette Janssen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11631)


A countable graph G is n-ordered if its vertices can be enumerated so each vertex has no more than n neighbours appearing earlier in the enumeration. Here we consider both deterministic and probabilistic methods to produce n-ordered countable graphs with universal adjacency properties. In the countably infinite case, we show that such universal adjacency properties imply the existence an independent 2-distinguishing labelling.


Graph evolution n-ordered graphs Graph distinguishing 


  1. 1.
    Abbasi, A., Hossain, L., Leydesdorff, L.: Betweenness centrality as a driver of preferential attachment in the evolution of research collaboration networks. J. Inf. 6(3), 403–412 (2012)Google Scholar
  2. 2.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Albertson, M.O., Collins, K.L.: Symmetry breaking in graphs. Electron. J. Comb. 3(1), 18 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Balachandran, N., Padinhatteeri, S.: Distinguishing chromatic number of random Cayley graphs. Discrete Math. 340(10), 2447–2455 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Barabási, A.L., Bonabeau, E.: Scale-free networks. Sci. Am. 288(5), 60–69 (2003)Google Scholar
  7. 7.
    Benzi, M., Klymko, C.: On the limiting behavior of parameter-dependent network centrality measures. SIAM J. Matrix Anal. Appl. 36(2), 686–706 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bonato, A.: The search for n-e.c. graphs. Contrib. Discret. Math. 4(1), 40–53 (2009)zbMATHGoogle Scholar
  9. 9.
    Bonato, A., Janssen, J., Wang, C.: The n-ordered graphs: a new graph class. J. Graph Theory 60(3), 204–218 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, London (2008). Scholar
  11. 11.
    Cameron, P.J.: The random graph revisited. Eur. Congr. Math. 1, 267–274 (2000)zbMATHGoogle Scholar
  12. 12.
    Cheng, C.T.: On computing the distinguishing and distinguishing chromatic numbers of interval graphs and other results. Discrete Math. 309(16), 5169–5182 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Choi, J.O., Hartke, S.G., Kaul, H.: Distinguishing chromatic number of Cartesian products of graphs. SIAM J. Discret. Math. 24(1), 82–100 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Collins, K.L., Trenk, A.N.: The distinguishing chromatic number. Electron. J. Comb. 13(1), 16 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Deijfen, M., Van Den Esker, H., Van Der Hofstad, R., Hooghiemstra, G.: A preferential attachment model with random initial degrees. Arkiv för Matematik 47(1), 41–72 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Hung. 14(3–4), 295–315 (1963)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Eschen, E.M., Hoàng, C.T., Sritharan, R., Stewart, L.: On the complexity of deciding whether the distinguishing chromatic number of a graph is at most two. Discrete Math. 311(6), 431–434 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Flaxman, A.D., Frieze, A.M., Vera, J.: A geometric preferential attachment model of networks. Internet Math. 3(2), 187–205 (2006)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Imrich, W., Klavžar, S., Trofimov, V.: Distinguishing infinite graphs. Electron. J. Comb. 14(1), R36 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Laflamme, C., Sauer, N., et al.: Distinguishing number of countable homogeneous relational structures. Electron. J. Comb. 17(1), R20 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    De Solla Price, D.: A general theory of bibliometric and other cumulative advantage processes. J. Am. Soc. Inf. Sci. 27(5), 292–306 (1976)Google Scholar
  22. 22.
    Ravasz, E., Barabási, A.L.: Hierarchical organization in complex networks. Phys. Rev. E 67(2), 026112 (2003)zbMATHGoogle Scholar
  23. 23.
    Russell, A., Sundaram, R.: A note on the asymptotics and computational complexity of graph distinguishability. Electron. J. Comb. 5(1), 23 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Telesford, Q.K., Joyce, K.E., Hayasaka, S., Burdette, J.H., Laurienti, P.J.: The ubiquity of small-world networks. Brain Connect. 1(5), 367–375 (2011)Google Scholar
  25. 25.
    Wang, Z., Scaglione, A., Thomas, R.J.: Generating statistically correct random topologies for testing smart grid communication and control networks. IEEE Trans. Smart Grid 1(1), 28–39 (2010)Google Scholar
  26. 26.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440 (1998)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada
  2. 2.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

Personalised recommendations