L(2,1)-labelings of the edge-multiplicity-paths-replacement of a graph

Abstract

An L(2,1)-labeling of a graph $$G$$G is an assignment of nonnegative integers to $$V(G)$$V(G) such that the difference between labels of adjacent vertices is at least $$2$$2, and the difference between labels of vertices that are distance two apart is at least 1. The span of an L(2,1)-labeling of a graph $$G$$G is the difference between the maximum and minimum integers used by it. The minimum span of an L(2,1)-labeling of $$G$$G is denoted by $$\lambda (G)$$?(G). This paper focuses on L(2,1)-labelings-number of the edge-multiplicity-paths-replacement $$G(rP_{k})$$G(rPk) of a graph $$G$$G. In this paper, we obtain that $$ r\Delta +1 \le \lambda (G(rP_{5}))\le r\Delta +2$$rΔ+1≤?(G(rP5))≤rΔ+2, $$\lambda (G(rP_{k}))= r\Delta +1$$?(G(rPk))=rΔ+1 for $$k\ge 6$$k?6; and $$\lambda (G(rP_{4}))\le (\Delta +1)r+1$$?(G(rP4))≤(Δ+1)r+1, $$\lambda (G(rP_{3}))\le (\Delta +1)r+\Delta $$?(G(rP3))≤(Δ+1)r+Δ for any graph $$G$$G with maximum degree $$\Delta $$Δ. And the L(2,1)-labelings-numbers of the edge-multiplicity-paths-replacement $$G(rP_{k})$$G(rPk) are completely determined for $$1\le \Delta \le 2$$1≤Δ≤2. And we show that the class of graphs $$G(rP_{k})$$G(rPk) with $$k\ge 3 $$k?3 satisfies the conjecture: $$\lambda ^{T}_{2}(G)\le \Delta +2$$?2T(G)≤Δ+2 by Havet and Yu (Technical Report 4650, 2002).