Let G(V,E) be a simple graph with m edges. For a given integer k, a k-shifted antimagic labeling is a bijection f:E(G)→{k+1,k+2,…,k+m} such that all vertices have different vertex-sums, where the vertex-sum of a vertex v is the total of the labels assigned to the edges incident to v. A graph G is {\it k-shifted antimagic} if it admits a k-shifted antimagic labeling. For the special case when k=0, a 0-shifted antimagic labeling is known as {\it antimagic labeling}; and G is {\it antimagic} if it admits an antimagic labeling. A spider is a tree with exactly one vertex of degree greater than two. A spider forest is a graph where each component is a spider. In this article, we prove that certain spider forests are k-shifted antimagic for all k≥0. In addition, we show that for a spider forest G with m edges, there exists a positive integer k0

The routing number $rt(G)$ of a connected graph G is the minimum integer r so that every permutation of vertices can be routed in r steps by swapping the ends of disjoint edges. In this paper, we study the routing numbers of cycles, complete bipartite graphs, and hypercubes. We prove that $rt(C_n)=n-1$ (for $n\geq3$) and for $s\geq t$, $rt(K_{s,t})=\lfloor\frac{3s}{2t}\rfloor+O(1)$. We also prove $n+1\leq rt(Q_n)\leq2n-2$ for $n\geq3$. The lower bound $rt(Q_n)\geq n+1$ was previously conjectured by Alon, Chung, and Graham [SIAM J. Discrete Math., 7 (1994), pp. 513–530]. A variation, called fractional routing number, is also considered in this paper.