Egalitarian Edge Orderings of Complete Graphs

For a consecutive ordering of the edges of a graph G= (V, E) , the point sum of a vertex is the sum of the indices of edges incident with that vertex. Motivated by questions of balancing accesses in data placements in the presence of popularity rankings, an edge ordering is egalitarian when all point sums are equal, and almost egalitarian when two point sums differ by at most 1. It is established herein that complete graphs on n vertices admit an egalitarian edge ordering when n≡1,2,3(mod4) and n∉ { 3 , 5 } , or an almost egalitarian edge ordering when n≡0(mod4) and n≠ 4.