# 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)
• 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.

#### Acknowledgement

Supported by : University of Malaya

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