Labeling the rr-path with a condition at distance two

For integer r≥2r≥2, the infinite rr-path P∞(r)P∞(r) is the graph on vertices …v−3,v−2,v−1,v0,v1,v2,v3……v−3,v−2,v−1,v0,v1,v2,v3… such that vsvs is adjacent to vtvt if and only if |s−t|≤r−1|s−t|≤r−1. The rr-path on nn vertices is the subgraph of P∞(r)P∞(r) induced by vertices v0,v1,v2,…,vn−1v0,v1,v2,…,vn−1. For non-negative reals x1x1 and x2x2, a λx1,x2λx1,x2-labeling of a simple graph GG is an assignment of non-negative reals to the vertices of GG such that adjacent vertices receive reals that differ by at least x1x1, vertices at distance two receive reals that differ by at least x2x2, and the absolute difference between the largest and smallest assigned reals is minimized. With λx1,x2(G)λx1,x2(G) denoting that minimum difference, we derive λx1,x2(Pn(r))λx1,x2(Pn(r)) for r≥3r≥3, 1≤n≤∞1≤n≤∞, and x1x2∈[2,∞]. For x1x2∈[1,2], we obtain upper bounds on λx1,x2(P∞(r))λx1,x2(P∞(r)) and use them to give λx1,x2(P∞(r))λx1,x2(P∞(r)) for r≥5r≥5 and x1x2∈[1,2r−22r−3]⋃[43,2]. We also determine λx1,x2(P∞(3))λx1,x2(P∞(3)) and λx1,x2(P∞(4))λx1,x2(P∞(4)) for all x1x2∈[1,2].