Cyclic Matching Sequencibility of Graphs
  We define the cyclic matching sequencibility of a graph to be the largest integer $d$ such that there exists a cyclic ordering of its edges so that every $d$ consecutive edges in the cyclic ordering form a matching. We show that the cyclic matching sequencibility of $K_{2m}$ and $K_{2m+1}$ equal $m-1$. 
Richard A. Brualdi, Kathleen P. Kiernan, Seth A. Meyer, Michael w.   Schroeder
math.CO