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.     

Maximum transversal in partial Latin squares and rainbow matchings

In a partial Latin square P a set of distinct entries, such that no two of which are in the same row or column is called a transversal. By the size of a transversal T, we mean the number of its entries. We define a duplex to be a partial Latin square of order n containing 2n entries such that exactly two entries lie in each row and column and each of n symbols occurs exactly twice. We show that determining the maximum size of a transversal in a given duplex is an NP-complete problem. This problem relates to independent sets in certain subfamilies of cubic graphs. Generalizing the concept of transversals in edge coloring of graphs we are led to introduce the concept of rainbow matching. We show that if each color appears at most twice then it is a polynomial time problem to know whether there exists a rainbow matching of size at least ⌊n/2⌋-t for each fixed t, where n is the order of the graph. As an application we show that for any fixed t, there is a polynomial time algorithm which decides whether α(G)?n-t, for any graph G on 2n vertices containing a perfect matching. At the end we mention some other applications of rainbow matching.     

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n-1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.     

Edge-deletable IM-extendable graphs with minimum number of edges

A graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. For a nonnegative integer k, a graph G is called a k-edge-deletable IM-extendable graph, if, for every F⊆E(G) with |F|=k, G−F is IM-extendable. In this paper, we characterize the k-edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer k, if G is ak-edge-deletable IM-extendable graph on 2n vertices, then |E(G)|≥(k+2)n; furthermore, the equality holds if and only if either G≅K_k+2,k+2, or k=4r−2 for some integer r≥3 and G≅C₅[N_2r], where N_2r is the empty graph on 2r vertices and C₅[N_2r] is the graph obtained from C₅ by replacing each vertex with a graph isomorphic to N_2r.     

Minimal rankings and the arank number of a path

Given a graph G, a function f:V(G)→{1,2,…,k} is a k-ranking of G if f(u)=f(v) implies every u-v path contains a vertex w such that f(w)>f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψ_r(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine ψ_r(P_n), a problem posed by Laskar and Pillone in 2000.     

Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v−1) off-diagonal cells can be resolved into v−1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v−2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.     

A multiply intersecting Erd?s-Ko-Rado theorem — The principal case

Let m(n,k,r,t) be the maximum size of satisfying |F₁∩?∩F_r|≥t for all F₁,…,F_r∈F. We prove that for every p∈(0,1) there is some r₀ such that, for all r>r₀ and all t with 1≤t≤⌊(p^1−r−p)/(1−p)⌋−r, there exists n₀ so that if n>n₀ and p=k/n, then . The upper bound for t is tight for fixed p and r.     

Linkability in iterated line graphs

We prove that for every graph H with the minimum degree δ?5, the third iterated line graph L³(H) of H contains as a minor. Using this fact we prove that if G is a connected graph distinct from a path, then there is a number k_G such that for every i?k_G the i-iterated line graph of G is -linked. Since the degree of Lⁱ(G) is even, the result is best possible.     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

On (s,t)-supereulerian graphs in locally highly connected graphs

Given two nonnegative integers s and t, a graph G is (s,t)-supereulerian if for any disjoint sets X,Y⊂E(G) with |X|≤s and |Y|≤t, there is a spanning eulerian subgraph H of G that contains X and avoids Y. We prove that if G is connected and locally k-edge-connected, then G is (s,t)-supereulerian, for any pair of nonnegative integers s and t with s+t≤k−1. We further show that if s+t≤k and G is a connected, locally k-edge-connected graph, then for any disjoint sets X,Y⊂E(G) with |X|≤s and |Y≤t, there is a spanning eulerian subgraph H that contains X and avoids Y, if and only if G−Y is not contractible to K₂ or to K_2,l with l odd.     

Graphic sequences with a realization containing a generalized friendship graph

Gould, Jacobson and Lehel [R.J. Gould, M.S. Jacobson, J. Lehel, Potentially G-graphical degree sequences, in: Y. Alavi, et al. (Eds.), Combinatorics, Graph Theory and Algorithms, vol. I, New Issues Press, Kalamazoo, MI, 1999, pp. 451-460] considered a variation of the classical Turán-type extremal problems as follows: for any simple graph H, determine the smallest even integer σ(H,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂+?+d_n≥σ(H,n) has a realization G containing H as a subgraph. Let F_t,r,k denote the generalized friendship graph on kt−kr+r vertices, that is, the graph of k copies of K_t meeting in a common r set, where K_t is the complete graph on t vertices and 0≤r≤t. In this paper, we determine σ(F_t,r,k,n) for k≥2, t≥3, 1≤r≤t−2 and n sufficiently large.     

Optimal interleaving schemes for correcting two-dimensional cluster errors

We present an elementary theory of optimal interleaving schemes for correcting cluster errors in two-dimensional digital data. It is assumed that each data page contains a fixed number of, say n, codewords with each codeword consisting of m code symbols and capable of correcting a single random error (or erasure). The goal is to interleave the codewords in the m×n array such that different symbols from each codeword are separated as much as possible, and consequently, an arbitrary error burst with size up to t can be corrected for the largest possible value of t. We show that, for any given m, n, the maximum possible interleaving distance, or equivalently, the largest size of correctable error bursts in an m×n array, is given by if n?⌈m²/2⌉, and t=m+⌊(n-⌈m²/2⌉)/m⌋ if n?⌈m²/2⌉. Furthermore, we develop a simple cyclic shifting algorithm that can provide a systematic construction of an m×n optimal interleaving array for arbitrary m and n. This extends important earlier work on the complementary problem of constructing interleaving arrays that, given the burst size t, minimize the interleaving degree, that is, the number of different codewords in a 2-D (or 3-D) array such that any error burst with given size t can be corrected. Our interleaving scheme thus provides the maximum burst error correcting power without requiring prior knowledge of the size or shape of an error burst.     

Degree conditions for group connectivity

Let G be a 2-edge-connected simple graph on n≥13 vertices and A an (additive) abelian group with |A|≥4. In this paper, we prove that if for every uv∉E(G), max{d(u),d(v)}≥n/4, then either G is A-connected or G can be reduced to one of K_2,3,C₄ and C₅ by repeatedly contracting proper A-connected subgraphs, where C_k is a cycle of length k. We also show that the bound n≥13 is the best possible.     

The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n^0.5−?) for any ?>0 unless NP=ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n^0.5−?) for any ?>0 unless NP=ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3.     

The existence of regular self-complementary 3-uniform hypergraphs

A k-uniform hypergraph with vertex set V and edge set E is called t-subset-regular if every t-element subset of V lies in the same number of elements of E. In this paper we show that a 1-subset-regular self-complementary 3-uniform hypergraph with n vertices exists if and only if n≥5 and n is congruent to 1 or 2 modulo 4.     

Relative difference sets fixed by inversion (III)—Cocycle theoretical approach

In this paper, we give a characterization of a group G which contains a semiregular relative difference set R relative to a central subgroup N containing the commutator subgroup [G,G] of G such that 1∈R and rRr=R for all r∈R. In particular, these relative difference sets are fixed by inversion and inner automorphisms of the group are multipliers. We also present a construction of such relative difference sets.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.