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ON THE MINIMUM ORDER OF 4-LAZY COPS-WIN GRAPHS

  • Sim, Kai An (Foundation, Study and Language Institute (FSLI) University of Reading Malaysia Persiaran Graduan Kota Ilmu) ;
  • Tan, Ta Sheng (Institute of Mathematical Sciences University of Malaya) ;
  • Wong, Kok Bin (Institute of Mathematical Sciences University of Malaya)
  • Received : 2017.10.30
  • Accepted : 2018.10.25
  • Published : 2018.11.30

Abstract

We consider the minimum order of a graph G with a given lazy cop number $c_L(G)$. Sullivan, Townsend and Werzanski [7] showed that the minimum order of a connected graph with lazy cop number 3 is 9 and $k_3{\square}k_3$ is the unique graph on nine vertices which requires three lazy cops. They conjectured that for a graph G on n vertices with ${\Delta}(G){\geq}n-k^2$, $c_L(G){\leq}k$. We proved that the conjecture is true for k = 4. Furthermore, we showed that the Petersen graph is the unique connected graph G on 10 vertices with ${\Delta}(G){\leq}3$ having lazy cop number 3 and the minimum order of a connected graph with lazy cop number 4 is 16.

Keywords

Acknowledgement

Supported by : University of Malaya

References

  1. M. Aigner and M. Fromme, A game of cops and robbers, Discrete Appl. Math. 8 (1984), no. 1, 1-11. https://doi.org/10.1016/0166-218X(84)90073-8
  2. W. Baird, A. Beveridge, A. Bonato, P. Codenotti, A. Maurer, J. Mccauley, and S. Valeya, On the minimum order of k-cop-win graphs, Contrib. Discrete Math. 9 (2014), no. 1, 70-84.
  3. A. Bonato, E. Chiniforooshan, and P. Pralat, Cops and Robbers from a distance, Theoret. Comput. Sci. 411 (2010), no. 43, 3834-3844. https://doi.org/10.1016/j.tcs.2010.07.003
  4. R. Nowakowski and P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983), no. 2-3, 235-239. https://doi.org/10.1016/0012-365X(83)90160-7
  5. D. Offner and K. Ojakian, Variations of cops and robber on the hypercube, Australas. J. Combin. 59 (2014), 229-250.
  6. A. Quilliot, Jeux et pointes fixes sur les graphes. These de 3eme cycle, Universite de Paris VI (1978), 131-145.
  7. B. W. Sullivan, N. Townsend, and M. Werzanski, The 3 ${\times}$ 3 rooks graph (K32K3) is the unique smallest graph with lazy cop number 3, arXiv: 1606.08485(2016).
  8. B. W. Sullivan, N. Townsend, and M. Werzanski, An introduction to lazy cops and robbers on graphs, College Math. J. 48 (2017), no. 5, 322-333. https://doi.org/10.4169/college.math.j.48.5.322