2 research outputs found

    Radio numbers for generalized prism graphs

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    A radio labeling is an assignment c:V(G)→Nc:V(G) \rightarrow \textbf{N} such that every distinct pair of vertices u,vu,v satisfies the inequality d(u,v)+|c(u)-c(v)|\geq \diam(G)+1. The span of a radio labeling is the maximum value. The radio number of GG, rn(G)rn(G), is the minimum span over all radio labelings of GG. Generalized prism graphs, denoted Zn,sZ_{n,s}, s≥1s \geq 1, n≥sn\geq s, have vertex set {(i,j) ∣ i=1,2andj=1,...,n}\{(i,j)\,|\, i=1,2 \text{and} j=1,...,n\} and edge set {((i,j),(i,j±1))}∪{((1,i),(2,i+σ)) ∣ σ=−⌊s−12⌋ …,0,…,⌊s2⌋}\{((i,j),(i,j \pm 1))\} \cup \{((1,i),(2,i+\sigma))\,|\,\sigma=-\left\lfloor\frac{s-1}{2}\right\rfloor\,\ldots,0,\ldots,\left\lfloor\frac{s}{2}\right\rfloor\}. In this paper we determine the radio number of Zn,sZ_{n,s} for s=1,2s=1,2 and 33. In the process we develop techniques that are likely to be of use in determining radio numbers of other families of graphs.Comment: To appear in Discussiones Mathematicae Graph Theory. 16 pages, 1 figur

    Radio Labelings of Distance Graphs

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    A radio kk-labeling of a connected graph GG is an assignment cc of non negative integers to the vertices of GG such that ∣c(x)−c(y)∣≥k+1−d(x,y),|c(x) - c(y)| \geq k+1 - d(x,y), for any two vertices xx and yy, x≠yx\ne y, where d(x,y)d(x,y) is the distance between xx and yy in GG. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set Z\Z of integers as vertex set and in which two distinct vertices i,j∈Zi, j \in \Z are adjacent if and only if ∣i−j∣∈D|i - j| \in D.Comment: 14 page
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