Embedding complete binary trees into star networks

Abstract. Star networks have been proposed as a possible interconnection network for massively parallel computers. In this paper we investigate embeddings of complete binary trees into star networks. Let G and H be two networks represented by simple undirected graphs. An embedding of G into H is an injective mapping f from the vertices of G into the vertices of H. The dilation of the embedding is the maximum distance between f(u), f(v) taken over all edges (u, v) of G. Low dilation embeddings of binary trees into star graphs correspond to efficient simulations of parallel algorithms that use the binary tree topology, on parallel computers interconnected with star networks. First, we give a construction of embeddings of dilation 1 of complete binary trees into n-dimensional star graphs. These trees are subgraphs of star graphs. Their height is fl(n log n), which is asymptotically optimal. Constructions of embeddings of complete binary trees of dilation 28 and 26 + 1, for 8 > 1, into star graphs are then given. The use of larger dilation allows embeddings of trees of greater height into star graphs. For example, the difference of the heights of the trees embedded with dilation 2 and 1 is greater than n/2. All these constructions can be modified to yield embeddings of dilation 1, and 26, for ~ > 1, of complete binary trees into pancake graphs. Our results show that massively parallel computers interconnected with star networks are well suited for efficient simulations of parallel algorithms with complete binary tree topology

Decomposition of Km,n(Km,n∗) into cycles (circuits) of length 2k

AbstractIn this paper we give necessary and sufficient conditions in order that Km,n (Km,n∗) admits a decomposition into 2k-cycles (2k-circuits). This answers conjectures of J. C. Bermond (Thesis, Paris XI (Orsay), 1975) and J. C. Bermond and V. Faber (J. Combinatorial Theory Ser. B 21 (1976), 146–155)

Decomposition of k∗n into circuits of odd length

AbstractIn this paper we prove that the complete symmetric directed graph with n vertices K∗n can be decomposed into directed circuits of length k, where k is an odd integer, if n ≡ 0 or 1 (mod k) and n ⩾ k

Embeddings of complete binary trees into grids and extended grids with total vertex-congestion 1

AbstractLet G and H be two simple, undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H, together with a mapping which assigns to each edge [u,v] of G a path between f(u) and f(v) in H. The grid M(r,s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j):0⩽i<r,0⩽j<s}, in which there is an edge between vertices (i,j) and (k,l) if either |i−k|=1 and j=l or i=k and |j−l|=1. The extended grid EM(r,s) is the graph whose vertex set is the set of pairs on nonnegative integers, {(i,j):0⩽i<r,0⩽j<s}, in which there is an edge between vertices (i,j) and (k,l) if and only if |i−k|⩽1 and |j−l|⩽1. In this paper, we give embeddings of complete binary trees into square grids and extended grids with total vertex-congestion 1, i.e., for any vertex x of the extended grid we have load(x)+vertex-congestion(x)⩽1. Depending on the parity of the height of the tree, the expansion of these embeddings is approaching 1.606 or 1.51 for grids, and 1.208 or 1.247 for extended grids

Number of arcs and cycles in digraphs

AbstractIn this article we determine the maximum number of arcs of a strong diagraph of order n, without a cycle of length at least n−k, for n⩾k2+2k+5; thus we partially solve a conjecture of Bermond, Germa, Heydemann and Sotteau.We give a conjecture concerning the maximum number of arcs of a strong diagraph of order n, of minimum half-degree r, without a cycle of length at least n−k, for large n. We prove it for r=2,k=1 or k=2

Digraphs without directed path of length two or three

AbstractWe characterize digraphs without any path of length two or of length three

On the existence of a matching orthogonal to a 2-factorization

AbstractThis note gives a partial answer to a problem posed by Brian Alspach in a recent issue of Discrete Mathematics. We show that if F1, F2,…,Fd is a 2-factorization of a 2d-regular graph G of order n⩾3.23d then G contains a d-matching with exactly one edge from each of F1, F2,…,Fd

About some cyclic properties in digraphs

AbstractIn this article we are concerned with digraphs in which any two vertices are on a common cycle. For example, we prove that, in a strong digraph of order n and half degrees at least 2 with at least n2 − 5n + 15 arcs, any two vertices are on a common cycle. We also consider related properties and give sufficient conditions on half degrees and the number of arcs to insure these properties. In particular, we show that every digraph of order n with half degrees at least r and with at least n2 − rn + r2 arcs is 2-linked