Low diameter graph decompositions

Adecomposition of a graphG=(V,E) is a partition of the vertex set into subsets (calledblocks). Thediameter of a decomposition is the leastd such that any two vertices belonging to the same connected component of a block are at distance d. In this paper we prove (nearly best possible) statements, of the form: Anyn-vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [4] it was shown that every graph has a decomposition into at mosts(n) blocks of diameter at mosts(n) for . Using a technique of Awerbuch [3] and Awerbuch and Peleg [5], we improve this result by showing that every graph has a decomposition of diameterO (logn) intoO(logn) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in timeO(log² n). The construction can be parameterized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible, for two families of graphs: the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner's lemma for the first class and Tucker's lemma for the second.A preliminary version of this paper appeared as Decomposing Graphs into Regions of Small Diameter in Proc. 2nd ACM-SIAM Symposium on Discrete Algorithms (1991) 321-330.This work was supported in part by NSF grant DMS87-03541 and by a grant from the Israel Academy of Science.This work was supported in part by NSF grant DMS87-03541 and CCR89-11388.     

Multigraded commutative algebra of graph decompositions

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products. Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K ₄-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.     

Balanced decompositions of a signed graph

The balanced decomposition number (b.d.n.) δ₀(Σ) of a signed graph Σ is the smallest number of balanced subsets into which its edges can be partitioned. (A special case is decomposition of a graph into bipartite subgraphs.) The connected b.d.n. δ₁(Σ) is the same, but the subsets must also be connected. The balanced partition number (b.p.n.) π₀(Σ) is the smallest size of a partition of the vertices into subsets inducing balanced subgraphs; the connected b.p.n. π₁(Σ) is similar but the induced subgraphs must be connected. We use signed graph coloring theory to prove that δ₀ = 1 + ⌈log₂ π₀⌉ and that δ₀ is analogous to a critical exponent in the sense of Crapo and Rota. We deduce bounds on δ₀ and values for special signed graphs. We show that δ₀ is computationally about as complex as the chromatic number. We prove that, for a complete signed graph, δ₁ = δ₀; more strongly, with three exceptions a minimal balanced decomposition exists into connected and spanning edge sets. And we show that, of δ₁ and 1 + ⌈log₂ π₁⌉, neither is always at least as large as the other.     

An existence theory for loopy graph decompositions

Let v≥k≥1 and λ≥0 be integers. Recall that a (v, k, λ) block design is a collection ??of k‐subsets of a v‐set X in which every unordered pair of elements in X is contained in exactly λ of the subsets in ??. Now let G be a graph with no multiple edges. A (v, G, λ) graph design is a collection ??of subgraphs, each isomoprhic to G, of the complete graph K_v such that each edge of K_v appears in exactly λof the subgraphs in ??. A famous result of Wilson states that for a fixed simple graph G and integer λ, there exists a (v, G, λ) graph design for all sufficiently large integers v satisfying certain necessary conditions. Here, we extend this result to include the case of loops in G. As a consequence, we obtain the asymptotic existence of equireplicate graph designs. Applications of the equireplicate condition are given. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:280‐289, 2011     

On Rosa-type labelings and cyclic graph decompositions

A labeling (or valuation) of a graph G is an assignment of integers to the vertices of G subject to certain conditions. A hierarchy of graph labelings was introduced by Rosa in the late 1960s. Rosa showed that certain basic labelings of a graph G with n edges yielded cyclic G-decompositions of K _2n+1 while other stricter labelings yielded cyclic G-decompositions of K _2nx+1 for all natural numbers x. Rosa-type labelings are labelings with applications to cyclic graph decompositions. We survey various Rosa-type labelings and summarize some of the related results. (Communicated by Peter Horák)     

On complementary decompositions of the complete graph

We consider decompositionsK _n H, whereH is eitherP ₃ (the path with 3 edges) or the complete bipartite graphK _{1, 3}, with the property that upon taking the complement of each graph in the decomposition one obtains a new decompositionK _n H ^c .Research supported in part by an NSERC postgraduate Scholarship.     

Directed star decompositions of the complete directed graph

An (s, t)-directed star is a directed graph with s + t + 1 vertices and s + t arcs; s vertices have indegree zero and outdegree one, t have indegree one and outdegree zero, and one has indegree s and outdegree t. An (s, t)-directed star decomposition is a partition of the arcs of a complete directed graph of order n into (s, t)-directed starsx. We establish necessary and sufficient conditions on s, t, and n for an (s, t)-directed star decomposition of order n to exist.     

Symmetric Hamilton cycle decompositions of the complete graph

We construct a new symmetric Hamilton cycle decomposition of the complete graph K_n for odd n > 7. © 2003 Wiley Periodicals, Inc.     

Perfect factors in the de Bruijn graph

A Perfect Factor in the de Bruijn graph is a collection of disjoint cycles of a fixed period with the property that each vertex of the graph lies on exactly one cycle. Alternatively a Perfect Factor is a set of periodic sequences of a fixed period such that for some fixedu, every possibleu-tuple of consecutive symbols occurs exactly once as a sub-sequence of a sequence of the set. As well as being of interest in their own right, Perfect Factors are fundamental in the construction of Perfect Maps in two dimensions. In this paper, necessary conditions on the period for the existence of Perfect Factors are given. These conditions are shown to be sufficient in the case where the size of the alphabet is a power of a prime. We also consider the construction of Perfect Factors over generalc-ary alphabets. Our results are then applied to obtain a large new class of non-binary Perfect Maps.Work carried out while at Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, U.K. Funded by SERC CASE award No. 90C/11574.     

On cyclic decompositions of the complete graph into the 2-regular graphs

The symbol C(m₁ ⁿ ₁m₂ ⁿ ₂...m_s ⁿ _s) denotes a 2-regular graph consisting ofn _i cycles of lengthm _i , i=1, 2,…,s. In this paper, we give some construction methods of cyclic(K _v ,G)-designs, and prove that there exists a cyclic(K _v , G)-design whenG=C((4m ₁) ⁿ ₁(4m ₂) ⁿ ₂...(4m _s ) ⁿ _s andv ≡ 1 (mod 2¦G¦).     

On perfect Γ‐decompositions of the complete graph

Generalizing the well‐known concept of an i‐perfect cycle system, Pasotti [Pasotti, in press, Australas J Combin] defined a Γ‐decomposition (Γ‐factorization) of a complete graph K_v to be i‐perfect if for every edge [x, y] of K_v there is exactly one block of the decomposition (factor of the factorization) in which x and y have distance i. In particular, a Γ‐decomposition (Γ‐factorization) of K_v that is i‐perfect for any i not exceeding the diameter of a connected graph Γ will be said a Steiner (Kirkman) Γ‐system of order v. In this article we first observe that as a consequence of the deep theory on decompositions of edge‐colored graphs developed by Lamken and Wilson [Lamken and Wilson, 2000, J Combin Theory Ser A 89, 149–200], there are infinitely many values of v for which there exists an i‐perfect Γ‐decomposition of K_v provided that Γ is an i‐equidistance graph, namely a graph such that the number of pairs of vertices at distance i is equal to the number of its edges. Then we give some concrete direct constructions for elementary abelian Steiner Γ‐systems with Γ the wheel on 8 vertices or a circulant graph, and for elementary abelian 2‐perfect cube‐factorizations. We also present some recursive constructions and some results on 2‐transitive Kirkman Γ‐systems. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 197–209, 2009     

Perfect matchings of a graph associated with a binary de Bruijn digraph

Let B_n be the binary de Bruijn digraph of order n and W the quotient set of the set of vertices of B_n with respect to the equivalence relation of rotation. Let G be the graph which has W as the set of vertices and in which two elements C and H are adjacent when there exist a vertex v of C and a vertex u of H such that (v,u) is an arc of B_n. Recently the problem of establishing whether the graph G has a perfect matching was posed. In this paper we answer in the positive to this problem in the case of n odd.     

Rainbow decompositions

A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson's Theorem. For every integer there exists an so that for all , if

then every properly edge-colored contains pairwise edge-disjoint rainbow copies of .

Our proof uses, as a main ingredient, a double application of the probabilistic method.