Self-complementary graph decompositions

Abstract

A complementary decomposition of XK n into a graph G is an edge-disjoint decomposition of lK n into copies of G such that if each copy H of G is replaced by its complement in V(H) then the result is an edge-disjoint decomposition of XK n into copies of G° ; it is a selfcomplementary decomposition if G = Gf . The spectrum for the last self-complementary graph on at most 7 vertices is found.

On large sets ofPk-decompositions

“…It is well known that if there exists a ðv; G; Þ-GD, then v ! n vðv À 1Þ 0 ðmod 2eÞ ðv À 1Þ 0 ðmod dÞ : where n ¼ jVðGÞj; e ¼ jEðGÞj; and d is the greatest common divisor of all the degrees of the vertices in G. For some simple graphs, such as path, star, cycle, k-cube, and the graphs with at most five vertices, the existence of their graph designs has been solved (see [1][2][3][4][5][6][7][8], [10][11]). Especially, for the path P k of k vertices, we have the following result.…”

Large set of P3‐decompositions

“…where d is the greatest common divisor of the degrees of the vertices of G. For some simple graphs, such as path, star, cycle, k-cube and the graphs with at most five vertices, the existence of these graph designs has been solved (see [1][2][3][4][5][6][7][8][9]). Especially, for the path P k of k vertices, we have the following result.…”