Distinguishing Cartesian Products of Countable Graphs

Ehsan Estaji 1 , Wilfried Imrich 2 , Rafał Kalinowski 3 , Monika Pilśniak 3 ,  and Thomas Tucker 4
  • 1 Hakim Sabzevari University, Sabzevar, Iran (Islamic Republic of)
  • 2 Montanuniversität Leoben, A-8700 Leoben, Austria
  • 3 AGH University, Department of Discrete Mathematics, 30-059 Krakow, Poland
  • 4 Colgate University, Hamilton NY 13346, United States of America

Abstract

The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] M.O. Albertson, Distinguishing Cartesian powers of graphs, Electron. J. Combin. 12 (2005) #N17.

  • [2] M.O. Albertson and K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996) #R18.

  • [3] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs (Second Edition), (Taylor & Francis Group, 2011).

  • [4] W. Imrich, Automorphismen und das kartesische Produkt von Graphen, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 177 (1969) 203–214.

  • [5] W. Imrich, Über das schwache kartesische Produkt von Graphen, J. Combin. Theory Ser. B 11 (1971) 1–16. doi:10.1016/0095-8956(71)90008-6

  • [6] W. Imrich, J. Jerebic and S. Klavžar, The distinguishing number of Cartesian products of complete graphs, European J. Combin. 29 (2008) 922–929. doi:10.1016/j.ejc.2007.11.018

  • [7] W. Imrich and S. Klavžar, Distinguishing Cartesian powers of graphs, J. Graph Theory 53 (2006) 250–260. doi:10.1002/jgt.20190

  • [8] W. Imrich, S. Klavžar and V. Trofimov, Distinguishing infinite graphs, Electron. J. Combin. 14 (2007) #R36.

  • [9] S. Klavžar and X. Zhu, Cartesian powers of graphs can be distinguished by two labels, European J. Combin. 28 (2007) 303–310. doi:10.1016/j.ejc.2005.07.001

  • [10] D.J. Miller, The automorphism group of a product of graphs, Proc. Amer. Math. Soc. 25 (1970) 24–28. doi:10.1090/S0002-9939-1970-0262116-3

  • [11] D.J. Miller, Weak cartesian product of graphs, Colloq. Math. 21 (1970) 55–74.

  • [12] A. Russell and R. Sundaram, A note on the asymptotics and computational complexity of graph distinguishability, Electron. J. Combin. 5 (1998) #R2.

  • [13] G. Sabidussi, Graph multiplication, Math. Z. 72 (1959/1960) 446–457.

  • [14] S.M. Smith, T. Tucker and M.E. Watkins, Distinguishability of infinite groups and graphs, Electron. J. Combin. 19 (2012) #P27.

  • [15] V.G. Vizing, The Cartesian product of graphs, Vychisl. Sistemy 9 (1963) 30–43, in Russian.

OPEN ACCESS

Journal + Issues

Search