| Abstract: | For integer r≥2, the infinite r-path P∞(r) is the graph on vertices …v−3,v−2,v−1,v0,v1,v2,v3… such that vs is adjacent to vt if and only if |s−t|≤r−1. The r-path on n vertices is the subgraph of P∞(r) induced by vertices v0,v1,v2,…,vn−1. For non-negative reals x1 and x2, a λx1,x2-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 x1, vertices at distance two receive reals that differ by at least x2, and the absolute difference between the largest and smallest assigned reals is minimized. With λx1,x2(G) denoting that minimum difference, we derive λx1,x2(Pn(r)) for r≥3, 1≤n≤∞, and  . For  , we obtain upper bounds on λx1,x2(P∞(r)) and use them to give λx1,x2(P∞(r)) for r≥5 and  . We also determine λx1,x2(P∞(3)) and λx1,x2(P∞(4)) for all  . |
