Almost self-complementary factors of complete bipartite graphs

Dalibor Fronček
1997 Discrete Mathematics  
A complete bipartite graph without one edge, IKn, m, is called almost complete bipartite graph. A graph /~2n+l,2m+l that can be decomposed into two isomorphic factors with a given diameter d is called d-isodecomposable. We prove that/~zn+l, 2"+1 is d-isodecomposable only if d = 3, 4, 5, 6 or oo and completely determine all d-isodecomposable almost complete bipartite graphs for each diameter. For d = oo we, moreover, present all classes of possible disconnected factors. A factor F of a graph G =
more » ... G(V,E) is a subgraph of G having the same vertex set V. A decomposition of a graph G(V,E) into two factors FI(V, E1) and F2(V, E2) is a pair of factors such that E1 n E2 =0 and El U E2=E. A decomposition of G is called isomorphic if F1 ~ F2. An isomorphism qS:Fl ---~F2 is then also called a self-complementin9 isomorphism, or complementing isomorphism and the factors F1 and F2 the self-complementary factors with respect to G or simply the self-complementary factors. The diameter diam G of a connected graph G is the maximum of the set of distances distc(x, y) among all pairs of vertices of G. If G is disconnected, then diam G = ~xD. The order of a graph G is the number of vertices of G while the size of G is the number of its edges. For terms not defined here, see [1]. Kotzig and Rosa [9] and later Tomasta [11], Palumbiny [10] and Hic and Palumbiny [8] studied decompositions of complete graphs into isomorphic factors with a given diameter. Tomovfi [12] studied decompositions of complete bipartite graphs *
doi:10.1016/s0012-365x(96)00237-3 fatcat:o5ndnts2efdsvnky6u52hqcprm