Decompositions of complete multipartite graphs into selfcomplementary factors with finite diameters

For r 4 we determine the smallest number of vertices, .<71-( d), of complete that are decomposable into two isomorphic factors for a given finite diameter d. We also prove that for a ,d such graph exists for each order than gr( d). 1. INTRODUCTORY NOTES AND DEFINITIONS In this paper we study decompositions of finite complete multipartite graphs factors with prescribed diameter. A factor F of a graph is a subgraph of G having the same vertex set V. A decomposition of into two factors Fl (V, Ed and F2 (V, is a pair of factors such o and El U E2 E. A decomposition of G is called isomorphic if . An isomorphism <p . Fl F2 is then called a complementing permutation and the factors Fl and F2 the selfcomplementary factors with respect to G or simply the selfcomplementary factors. The diameter diam G of connected graph G is the maximum of the of distances y) among all of vertiees of G. If G then diam G 00. The order of a G is the number of vertices of G while the size of G is the number of its For terms not defined [1]. and A. Rosa [7] and later P. Tomasta [9], D. Palumbfny [8], and studied decompositions of complete graphs into isodiameter. E. Tomova [10] studied decompositions of into two factors with given diameters and determined all of diameters of sueh factors. T. Gangopadhyay [5] studied decompositiom; of (1" ~ 3) into two factors with given diameters and determined also all possible pairs of diameters of such factors. *Current address: 17. listopadu, 708 Technical University Ostrava, Department of Applied Mathematics, FEI, Ostrava, Czech Republic. E-Mail: [email protected] Austt~alasian JoUt~nal of Combinatorics 11(1996), pp.61-74 In this article we join both concepts. We study decompositions of complete T-partite graphs, for r' ;:::: 5 into two isomorphic factors with a given diameter (for T 2,3,4 see [3],[4]). We always assume that the number of vertices of an r-partite graph is at least T + 1, i.e. the graph is not a complete J{,. T. Gangopadhyay [5] proved that a complete r-partite graph for r ;:::: 3 decomposable into two factors with the same finite diameter d exists if and only if d 2,3,4 or 5. He also determined the smallest orders of such decomposable graphs. A complete T-partite graph is d-decomposable if it is decomposable into two fadors with the same finite diameter d. If we in addition require the factors to be mutually isomorphic, we say that the graph is d-isodecomposable. We also often say that a graph G is isodecomposable if it is d-isodecomposable for a finite diameter d which we do not determine specifically. We show that there are d-isodecomposable of the above mentioned diameters for any r slnallest decomposable graphs. 2. PRELIMINARY THEOREMS T'-partite graphs for each 5. In all cases we also present We denote a complete r-partite graph with r partite sets having ml, m2, ... , m, vertices, respectively, ]{mI,m2, .. ,m r • Or, if there are more having the same cardinality, we denote the complete graph having of cardinality ni for i 1,2, ... ,s by J( hI In this case we always suppose n I that kl + k2 + ... + ks T and ni i= nj for i= j. Let f,( d) denote the smallest number of vertices of a complete r-partite ddecomposable graph. If such a number does not then we define f,(d) 00. It is obvious that any d-isodecomposable complete T-partite graph J(mI ,m2, ... ,mr must have an even number of edges and hence the number of parts having odd cardinalities must be 0 or 1 (mod 4). A graph with this property as well as the corresponding T-tuple ml, m2, ... ,m, is called admissible. We can similarly introduce g,( d) as the smallest number of vertices of a complete d-isodecomposable T-partite graph. We also define g~ (d) as the smallest integer with the property that for any n ;:::: g~( d) there is a complete r-partite disodecomposable graph with n vertices. Finally, we define h,( d) as the smallest integer such that any admissible complete T-partite graph with at least h,( d) vertiees is d-isodecomposable. If such numbers do not exist, we again put g,(d) = 00, g;.( d) 00 or h,( d) = 00, respectively. It is obvious that f,(d) ~ g,(d) ~ g~(d) ~ h,(d). The first and last inequality can be in some cases sharp. For instance, Gangopadhyay [5] proved that f,(2) = r + 1, but we show that g,(2) = T + 1 only if T == 1 or 2(mod4) while g,(2) = r + 2 for r == O(mod4) and g,(2) = r + 3 for T == 3(mod4). The last inequality can be sharp as well: for r == O(mod4) it holds