Labeling the r-path with a condition at distance two

For integer r=2, the infinite r-path P"~(r) is the graph on vertices ...v"-"3,v"-"2,v"-"1,v"0,v"1,v"2,v"3... such that v"s is adjacent to v"t if and only if |s-t|@?r-1. The r-path on n vertices is the subgraph of P"~(r) induced by vertices v"0,v"1,v"2,...,v"n"-"1. For non-negative reals x"1 and x"2, a @l"x"""1","x"""2-labeling of a simple graph G is an assignment of non-negative reals to the vertices of G such that adjacent vertices receive reals that differ by at least x"1, vertices at distance two receive reals that differ by at least x"2, and the absolute difference between the largest and smallest assigned reals is minimized. With @l"x"""1","x"""2(G) denoting that minimum difference, we derive @l"x"""1","x"""2(P"n(r)) for r=3, 1@?n@?~, and x"1x"2@?[2,~]. For x"1x"2@?[1,2], we obtain upper bounds on @l"x"""1","x"""2(P"~(r)) and use them to give @l"x"""1","x"""2(P"~(r)) for r=5 and x"1x"2@?[1,2r-22r-3]@?[43,2]. We also determine @l"x"""1","x"""2(P"~(3)) and @l"x"""1","x"""2(P"~(4)) for all x"1x"2@?[1,2].