A theorem on the maximum number of disjoint steiner triple systems

We prove that D(2v + 1) ? v + 1 + D(v) for v > 3 where D(v) denotes the maximum number of pairwise disjoint Steiner triple systems of order v. Since D(v) ? v ? 2 it follows that for v > 3, D(2v + 1) = 2v ?1 whenever D(v) = v ? 2.     

The fine triangle intersections for maximum kite packings

In this paper, the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let Fin(v) = {(s,t) : a pair of maximum kite packings of order v intersecting in s blocks and s+t triangles}. Let Adm(v) = {(s, t) : s + t ≤ bv , s,t are non-negative integers}, where b v = v(v 1)/8 . It is established that Fin(v) = Adm(v)\{(bv-1, 0), (bv-1,1)} for any integer v ≡ 0, 1 (mod 8) and v ≥ 8; Fin(v) = Adm(v) for any integer v ≡ 2, 3, 4, 5, 6, 7 (mod 8) and v ≥ 4.     

On 2-designs

Denote by M_v the set of integers b for which there exists a 2-design (linear space) with v points and b lines. M_v is determined as accurately as possible. On one hand, it is shown for v > v₀ that M_v contains the interval [v + p + 1, v + p + q ? 1]. On the other hand for v of the form p² + p + 1 it is shown that the interval [v + 1, v + p ? 1] is disjoint from M_v; and if v > v₀ and p is of the form q² + q, then an additional interval [v + p + 1, v + p + q ? 1] is disjoint from M_v.     

Clique partitions of the cocktail party graph

Let T_v denote the complenent of a perfect matching in the complete graph on v vertices, v even, and let cp(T_v) be the minimum number of cliques needed to partition the edge-set of T_v. We prove that cp(T_v) ⩾ v for v⩾ 8 and give a design characterization of the cases where equality holds. We also show that, asymptotically, cp(T_v) ⩽ v log log v.     

The Ramsey number for hypergraph cycles I

Let C_n denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v₁,…,v_n and edges v₁v₂v₃, v₃v₄v₅, v₅v₆v₇,…,v_n-1v_nv₁. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C_n, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible.     

Accretion centers: A generalization of branch weight centroids

An ordered n-tuple (v_i1,v_i2,…,v_in) is called a sequential labelling of graph G if {v_i1,v_i2,…,v_in} = V(G) and the subgraph induced by {v_i1,v_i2,…, v_ij} is connected for 1≤j≤n. Let σ(v;G) denote the number of sequential labellings of G with v_i1=v. Vertex v is defined to be an accretion center of G if σ is maximized at v. This is shown to generalize the concept of a branch weight centroid of a tree since a vertex in a tree is an accretion center if and only if it is a centroid vertex. It is not, however, a generalization of the concept of a median since for a general graph an accretion center is not necessarily a vertex of minimum distance. A method for computing σ(v;G) based upon edge contractions is described.     

A generalization of a result of Sylvester's

Let v₁,…,v_n be vectors in Zⁿ with D = det(v₁,…,v_n) > 0. Let v_{n + 1} be in the cone generated by v₁,…,v_n and such that v₁,…,v_v, v_{n + 1} generate Zⁿ as a Z-module. There exists a unique “largest“ χ not expressible as a nonnegative integer combination of v₁,…,v_n, v_{n + 1} and χ = Dv_{n + 1} ? (v₁ + … v_n + v_{n + 1}).     

Halving Steiner 2-designs

A Steiner 2-design S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G on v vertices into K_ks. The obvious necessary condition of those orders v for which there exists a halvable S(2,k,v) is that v admits the existence of an S(2,k,v) with an even number of blocks. In this paper, we give an asymptotic solution for various block sizes. We prove that for any k?5 or any Mersenne prime k, there is a constant number v₀ such that if v>v₀ and v satisfies the above necessary condition, then there exists a halvable S(2,k,v). We also show that a halvable S(2,2ⁿ,v) exists for over a half of possible orders. Some recursive constructions generating infinitely many new halvable Steiner 2-designs are also presented.     

Elementary divisors of induced transformations on symmetry classes of tensors

Let V_χ(G) denote the symmetry class of tensors over the vector space V associated with the permutation group G and irreducible character χ. Write v₁*v₂*...*v_m for the decomposable symmetrized product of the indicated vectors (m=degG). If T is a linear operator on V, let K(T) denote the associated operator on V_χ(G), i.e., K(T)v₁*v₂*...*v_m=Tv₁*Tv₂*...*Tv_m. Denote by D(T) the derivation operator D(T)v₁*v₂*...*v_m=Tv₁*v₂...*v_m+v₁*Tv₂*v₃* ...*v_m+...+v₁*v₂*...*v_m–1*Tv_m. The article concerns the elementary divisors of K(T) and D(T).     

On large sets of disjoint steiner triple systems,VI

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that D(v) = v ? 2 holds for all v ≡ 1, 3 (mod 6) (v>7), except possibly v = 141, 283, 501, 789, 1501, 2365.     

On the maximum number of disjoint Steiner triple systems

Let D(v) be the maximum number of pairwise disjoint Steiner triple systems of order v. We prove that D(3v)≥2v+D(v) for every v ≡ 1 or 3 (mod 6), v≥3. As a corollary, we have D(3ⁿ)=3ⁿ-2 for every n≥1.     

Existence of r-self-orthogonal Latin squares

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the first one, we say that the first square is r-self-orthogonal, denoted by r-SOLS(v). It has been proved that for any integer v?28, there exists an r-SOLS(v) if and only if v?r?v² and r∉{v+1,v²-1}. In this paper, we give an almost complete solution for the existence of r-self-orthogonal Latin squares.     

Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation

vt−αvxxt−βvxx+γvx+fx(v)=φx(vx)+g(v)−αg(v)_xx     

Some characterizations of Krull domains

In this paper, we define the v-finiteness for a length function L_v on the set of all v-ideals of an integral domain R and show that R is a Krull domain if and only if every proper integral v-ideal of R has v-finite length and L_v((AB)_v)=L_v(A)+L_v(B) for every pair of proper integral v-ideals A and B in R. We also give Euclidean-like characterizations of factorial, Krull, and π-domains. Finally we define the notion of quasi-∗-invertibility and show that if every proper prime t-ideal of an integral domain R is quasi-t-invertible, then R is a Krull domain.     

A note on graphs whose neighborhoods are n-cycles

Let G be a graph, and let v be a vertex of G. We denote by N(v) the set of vertices of G which are adjacent to v, and by 〈N(v)〉 the subgraph of G induced by N(v). We call 〈N(v)〉 the neighborhood of v. In a paper of 1968, Agakishieva has, as one of her main theorems, the statement: “Graphs in which every neighborhood is an n-cycle exist if and only if 3?n?6.” It it is the object of this note to provide a list of counter examples to this statement.     

Geodeticity of the contour of chordal graphs

A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. Moreover, if no vertex in the whole graph V(G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V(G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.     

Interval-regular graphs

An interval-regular graph is a connected graph in which, for any two vertices u and v, the number of neighbours of u on all shortest (u, v)-paths equals d(u, v). It is proved that in an interval-regular graph the shortest (u, v)-paths induce a hypercube of dimension d(u, v), for any two vertices u and v. The products of complete graphs are characterized as interval-regular graphs satisfying some extra conditions. The extended odd graphs are introduced as critical example with respect to the results proved.     

On Harary index of graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. For a connected graph G=(V,E) and two nonadjacent vertices v_i and v_j in V(G) of G, recall that G+v_iv_j is the supergraph formed from G by adding an edge between vertices v_i and v_j. Denote the Harary index of G and G+v_iv_j by H(G) and H(G+v_iv_j), respectively. We obtain lower and upper bounds on H(G+v_iv_j)−H(G), and characterize the equality cases in those bounds. Finally, in this paper, we present some lower and upper bounds on the Harary index of graphs with different parameters, such as clique number and chromatic number, and characterize the extremal graphs at which the lower or upper bounds on the Harary index are attained.     

One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A _v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A _v . We show that for v ≠ ord _R , A _v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A _v ; v) is 1. We then give a criterion for R to be regular.