Routing Permutations on Graphs Via Matchings
SIAM Journal on Discrete Mathematics
Optimal Bounds for Matching Routing on Trees
SIAM Journal on Discrete Mathematics
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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.