A proof of Lindner's conjecture on embeddings of partial Steiner triple systems

Lindner's conjecture that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if and is proved. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 63–89, 2009     

Quasi‐embeddings of Steiner triple systems,or Steiner triple systems of different orders with maximum intersection

In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v ≡ 1,3 (mod 6), u < v < 2u + 1, we ask for the minimum r such that there exists a Steiner triple system such that some partial system can be completed to an STS , where |?| = r. In other words, in order to “quasi‐embed” an STS(u) into an STS(v), we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity (u(u ? 1)/6) ? r as the maximum intersection of an STS(u) and an STS(v) with u < v. We conjecture that the necessary minimum r = (v ? u) (2u + 1 ? v)/6 can be achieved, except when u = 6t + 1 and v = 6t + 3, in which case it is r = 3t for t ≠ 2, or r = 7 when t = 2. Using small examples and recursion, we solve the cases v ? u = 2 and 4, asymptotically solve the cases v ? u = 6, 8, and 10, and further show for given v ? u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v ? u). Some results are obtained for v close to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.     

Small Embeddings of Partial Steiner Triple Systems

It was proved in 2009 that any partial Steiner triple system of order u has an embedding of order v for each admissible . This result is best possible in the sense that, for each , there exists a partial Steiner triple system of order u that does not have an embedding of order v for any . Many partial Steiner triple systems do have embeddings of orders smaller than , but much less is known about when these embeddings exist. In this paper, we detail a method for constructing such embeddings. We use this method to show that each member of a wide class of partial Steiner triple systems has an embedding of order v for at least half (or nearly half) of the orders for which an embedding could exist. For some members of this class we are able to completely determine the set of all orders for which the member has an embedding.     

A S(3, {4, 6}, 18) with a subdesign S(3, 4, 8) does not exist

For , a S(t,K,v) design is a pair, , with |V| = v and a set of subsets of V such that each t‐subset of V is contained in a unique and for all . If , , , and is a S(t,K,u) design, then we say has a subdesign on U. We show that a S(3,{4,6},18) design with a subdesign S(3,4,8) does not exist. © 2007 Wiley Periodicals, Inc. J Combin Designs 17: 36–38, 2009     

An improvement on H design

An H(m, g, 4, 3) is a triple , where X is a set of mg points, is a partition of X into m disjoint sets of size g, and is a set of 4‐element transverses of , such that each 3‐element transverse of is contained in exactly one of them. Such a design was introduced by Hanani 2 , who used it to study Steiner quadruple systems. Mills showed that for , an H(m, g, 4, 3) exists if and only if mg is even and is divisible by 3, and that for , an H(5, g, 4, 3) exists if g is divisible by 4 or 6 10 . In this article, we shall show that an H(5, g, 4, 3) exists if g is even, and . © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 25–35, 2009     

Existence of 3‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly SQS (v)) is a pair (X, ), where X is a v‐element set and is a set of 4‐element subsets of X (called blocks or quadruples), such that each 3‐element subset of X is contained in a unique block of . The chromatic number of an SQS(v)(X, ) is the smallest m for which there is a map such that for all , where . The system (X, ) is equitably m‐chromatic if there is a proper coloring with minimal m for which the numbers differ from each other by at most 1. Linek and Mendelsohn showed that an equitably 3‐chromatic SQS(v) exists for v ≡ 4, 8, 10 (mod 12), v ≥ 16. In this article we show that an equitably 3‐chromatic SQS(v) exists for v ≡ 2 (mod 12) with v > 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 469–477, 2007     

The spectrum of equitable, 4‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly an SQS(v)) is a pair (X, ) with |X| = v and a set of quadruples taken from X such that every triple in X is in a unique quadruple in . Hanani [Canad J Math 12 (1960), 145–157] showed that an SQS(v) exists if and only if v is {admissible}, that is, v = 0,1 or v ≡ 2,4 (mod 6). Each SQS(v) has a chromatic number when considered as a 4‐uniform hypergraph. Here we show that a 4‐chromatic SQS(v) exists for all admissible v ≥ 20, and that no 4‐chromatic SQS(v) exists for v < 20. Each system we construct admits a proper 4‐coloring that is equitable, that is, any two color classes differ in size by at most one. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 369–392, 2007     

Coloring Cubic Graphs by Point‐Intransitive Steiner Triple Systems

An ‐coloring of a cubic graph G is an edge coloring of G by points of a Steiner triple system such that the colors of any three edges meeting at a vertex form a block of . A Steiner triple system that colors every simple cubic graph is said to be universal. It is known that every nontrivial point‐transitive Steiner triple system that is neither projective nor affine is universal. In this article, we present the following results.

相似文献   

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Giant components in Kronecker graphs

Let \begin{align*}n\in\mathbb{N}\end{align*}, 0 <α,β,γ< 1. Define the random Kronecker graph K(n,α,γ,β) to be the graph with vertex set \begin{align*}\mathbb{Z}_2^n\end{align*}, where the probability that u is adjacent to v is given by p_u,v =α ^{u ? v} γ^{( 1‐u )?( 1‐v )}β^{n‐ u ? v ‐( 1‐u )?( 1‐v )}. This model has been shown to obey several useful properties of real‐world networks. We establish the asymptotic size of the giant component in the random Kronecker graph.© 2011 Wiley Periodicals, Inc. Random Struct. Alg.,2011     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008     

Circular choosability via combinatorial Nullstellensatz

A p‐list assignment L of a graph G assigns to each vertex v of G a set of permissible colors. We say G is L‐(P, q)‐colorable if G has a (P, q)‐coloring h such that h(v) ? L(v) for each vertex v. The circular list chromatic number of a graph G is the infimum of those real numbers t for which the following holds: For any P, q, for any P‐list assignment L with , G is L‐(P, q)‐colorable. We prove that if G has an orientation D which has no odd directed cycles, and L is a P‐list assignment of G such that for each vertex v, , then G is L‐(P, q)‐colorable. This implies that if G is a bipartite graph, then , where is the maximum average degree of a subgraph of G. We further prove that if G is a connected bipartite graph which is not a tree, then . © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 190–204, 2008     

Phase transition for potentials of high‐dimensional wells

For a potential function that attains its global minimum value at two disjoint compact connected submanifolds N^± in , we discuss the asymptotics, as ? → 0, of minimizers u_? of the singular perturbed functional under suitable Dirichlet boundary data . In the expansion of E _? (u_?) with respect to , we identify the first‐order term by the area of the sharp interface between the two phases, an area‐minimizing hypersurface Γ, and the energy c of minimal connecting orbits between N⁺ and N^?, and the zeroth‐order term by the energy of minimizing harmonic maps into N^± both under the Dirichlet boundary condition on ?Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.     

Trinal Decompositions of Steiner Triple Systems into Triangles

It is well known that when or , there exists a Steiner triple system (STS) of order n decomposable into triangles (three pairwise intersecting triples whose intersection is empty). A triangle in an STS determines naturally two more triples: the triple of “vertices” , and the triple of “midpoints” . The number of these triples in both cases, that of “vertex” triples (inner) or that of “midpoint triples” (outer), equals one‐third of the number of triples in the STS. In this paper, we consider a new problem of trinal decompositions of an STS into triangles. In this problem, one asks for three distinct decompositions of an STS of order n into triangles such that the union of the three collections of inner triples (outer triples, respectively) from the three decompositions form the set of triples of an STS of the same order. These decompositions are called trinal inner and trinal outer decompositions, respectively. We settle the existence question for trinal inner decompositions completely, and for trinal outer decompositions with two possible exceptions.     

A list analogue of equitable coloring

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003     

Global and blow‐up solutions for non‐linear degenerate parabolic systems

In this paper the degenerate parabolic system u_t=u(u_xx+av). vt=v(v_xx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α₁=α₂) of solution are analysed. For , the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

Large Cross‐Free Sets in Steiner Triple Systems

A cross‐free set of size m in a Steiner triple system is three pairwise disjoint m‐element subsets such that no intersects all the three ‐s. We conjecture that for every admissible n there is an STS(n) with a cross‐free set of size which if true, is best possible. We prove this conjecture for the case , constructing an STS containing a cross‐free set of size 6k. We note that some of the 3‐bichromatic STSs, constructed by Colbourn, Dinitz, and Rosa, have cross‐free sets of size close to 6k (but cannot have size exactly 6k). The constructed STS shows that equality is possible for in the following result: in every 3‐coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic connected component of size at least (we conjecture that equality holds for every admissible n). The analog problem can be asked for r‐colorings as well, if and is a prime power, we show that the answer is the same as in case of complete graphs: in every r‐coloring of the blocks of any STS(n), there is a monochromatic connected component with at least points, and this is sharp for infinitely many n.     

Small degree out‐branchings

Using a suitable orientation, we give a short proof of a strengthening of a result of Czumaj and Strothmann 4 : Every 2‐edge‐connected graph G contains a spanning tree T with the property that for every vertex v. As an analogue of this result in the directed case, we prove that every 2‐arc‐strong digraph D has an out‐branching B such that . A corollary of this is that every k‐arc‐strong digraph D has an out‐branching B such that , where . We conjecture that in this case would be the right (and best possible) answer. If true, this would again imply a strengthening of a result from 4 concerning spanning trees with small degrees in k‐connected graphs when k ≥ 2. We prove that for acyclic digraphs the existence of an out‐branching satisfying prescribed bounds on the out‐degrees of each vertex can be checked in polynomial time. A corollary of this is that the existence of arc‐disjoint branchings , , where the first is an out‐branching rooted at s and the second an in‐branching rooted at t, can be checked in polynomial time for the class of acyclic digraphs © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 297–307, 2003     

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, , is a sequence of graphs, where is the edgeless graph on n vertices, and is the result of adding an edge to , uniformly distributed over all the missing edges. The authors show that in almost every graph process equals the minimal degree of as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to , its final value. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008