The Optimal Rubbling Number of Paths, Cycles, and Grids

Table 1

Bounds on optimal rubbling of .

	2j+1	2j+1	2j+2	2j+1	2j+1	2j+2	2j+1	2j+1	2j+2
	2(k−j)+1	2(k−j)	2(k−j)−1	2(k−j)+1	2(k−j)+1	2(k−j)	2(k−j)+2	2(k−j)+1	2(k−j)+1
Lower bound	2k+2	2k+1	2k+1	2k+2	2k+2	2k+2	2k+3	2k+2	2k+3
	2k+1	2k+1	2k+1	2k+2	2k+2	2k+2	2k+2	2k+2	2k+2