On the minimum order of 4-lazy cops-win graphs

• Main Authors: Sim, Kai An, Tan, Ta Sheng, Wong, Kok Bin
• Format: Article
• Published: Korean Mathematical Society 2018
• Subjects: Q Science (General); QA Mathematics
• Online Access: http://eprints.um.edu.my/20660/; http://pdf.medrang.co.kr/kms01/BKMS/55/BKMS-55-6-1667-1690.pdf

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 □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 ∆(G) ≥ n − k 2 , c L (G) ≤ 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 ∆(G) ≤ 3 having lazy cop number 3 and the minimum order of a connected graph with lazy cop number 4 is 16.