Slight improvements of the Singleton bound

The problem of determining A_q(n,d), the maximum cardinality of a q-ary code of length n with minimum distance at least d, is considered in some cases where corresponding MDS codes do not exist. Slight improvements of the Singleton bound are given, including A_q(q+2,q)?q³-3 if q is odd, A₅(7,5)?5³-4 and A₁₆(18,15)?18⁴-4.     

The Independence Number for De Bruijn networks and Kautz networks

Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by α_l,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that α_l,d−1(B(d,n))=dⁿ,α_l,d−1(K(d,n))=α_l,d(K(d,n))=dⁿ+dⁿ⁻¹ if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound α_l,d−1(B(d,n))≥d² if n≥3 and d≥5.     

On time-relaxed broadcasting networks

Given a communication network (modelled as a graph), a message is transmitted to at one vertex transmits to all other vertices in such a way that each message transmission takes one time unit and each vertex participates in at most one transmission to its neighbor per time step. We call this process broadcasting. For t≥0, let B_t(n) be the number of edges in the sparsest possible graph on n vertices in which broadcasting can be accomplished in ⌈log₂n⌉+t steps regardless of the originator. Shastri [A. Shastri, Time-relaxed broadcasting in communication networks, Discrete Applied mathematics 83 (1998) 263-278] conjectured that B₁(22)=24 and B₂(n)=n+1 for 25≤n≤29. In this paper, we show that B₁(22)=24, B₂(n)=n for 25≤n≤28 and 37≤n≤44, B₂(n)≤n+1 for 45≤n≤49, B₂(n)≤n+4 for 50≤n≤56, and B₃(n)=n for 55≤n≤64.     

Graph designs for the eight-edge five-vertex graphs

The existence of graph designs for the two nonisomorphic graphs on five vertices and eight edges is determined in the case of index one, with three possible exceptions in total. It is established that for the unique graph with vertex sequence (3, 3, 3, 3, 4), a graph design of order n exists exactly when and n≠16, with the possible exception of n=48. For the unique graph with vertex sequence (2,3,3,4,4), a graph design of order n exists exactly when , with the possible exceptions of n∈{32,48}.     

Improving the high order nonlinearity lower bound for Boolean functions with given algebraic immunity

Algebraic immunity is a recently introduced cryptographic parameter for Boolean functions used in stream ciphers. If pAI(f) and pAI(f⊕1) are the minimum degree of all annihilators of f and f⊕1 respectively, the algebraic immunity AI(f) is defined as the minimum of the two values. Several relations between the new parameter and old ones, like the degree, the r-th order nonlinearity and the weight of the Boolean function, have been proposed over the last few years.In this paper, we improve the existing lower bounds of the r-th order nonlinearity of a Boolean function f with given algebraic immunity. More precisely, we introduce the notion of complementary algebraic immunity defined as the maximum of pAI(f) and pAI(f⊕1). The value of can be computed as part of the calculation of AI(f), with no extra computational cost. We show that by taking advantage of all the available information from the computation of AI(f), that is both AI(f) and , the bound is tighter than all known lower bounds, where only the algebraic immunity AI(f) is used.     

Scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions.     

Ramsey numbers for a disjoint union of good graphs

We give the Ramsey number for a disjoint union of some G-good graphs versus a graph G generalizing the results of Stahl (1975) [5] and Baskoro et al. (2006) [1] and the previous result of the author Bielak (2009) [2]. Moreover, a family of G-good graphs with s(G)>1 is presented.     

Linear-time certifying algorithms for near-graphical sequences

A sequence 〈d₁,d₂,…,d_n〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose ith vertex has degree d_i, for 1≤i≤n. The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs.If H=(V,E) is a graph and g and f are integer-valued functions on the vertex set V, then a (g,f)-factor of H is a subgraph G=(V,F) of H whose degree at each vertex v∈V lies in the interval [g(v),f(v)]. Thus, a (0,1)-factor is just a matching of H and a (1, 1)-factor is a perfect matching of H. If H is complete then a (g,f)-factor realizes a degree sequence that is consistent with the sequence of intervals 〈[g(v₁),f(v₁)],[g(v₂),f(v₂)],…,[g(v_n),f(v_n)]〉.Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient (g,f)-factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient (g,f)-factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing (f,f)-factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös-Gallai characterization of (f,f)-factors in complete graphs) at no additional cost.     

A degree sum condition for graphs to be prism hamiltonian

Win, in 1975, and Jackson and Wormald, in 1990, found the best sufficient conditions on the degree sum of a graph to guarantee the properties of “having a k-tree” and “having a k-walk”, respectively. The property of “being prism hamiltonian” is an intermediate property between “having a 2-tree” and “having a 2-walk”. Thus, it is natural to ask what is the best degree sum condition for graphs to be prism hamiltonian. As an answer to this problem, in this paper, we show that a connected graph G of order n with σ₃(G)≥n is prism hamiltonian. The degree sum condition “σ₃(G)≥n” is best possible.     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).     

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.     

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.     

Constructions for cyclic Moebius ladder systems

J.A. Gallian has proved [J.A. Gallian, Labeling prisms and prism related graphs, Congr. Numer. 59 (1987) 89-100] that every cubic graph M_2k obtainable from a 2k-cycle by adding its k diameters (the so-called Moebius Ladder of order 2k) is graceful. Here, in the case of k even, we propose a new graceful labeling that besides being simpler than Gallian’s one is able to give, at the same time, a graceful labeling of the prism of order 2k. Most importantly in the case of k odd, namely in the bipartite case, we prove that M_2k also admits an α-labeling. This implies that there exists a cyclic decomposition of the complete graph K_6kt+1 into copies of M_2k for every pair of positive integers k and t with k odd.In some cases we are able to give such decompositions also when k is even. Apart from the case of t=1 that is an obvious consequence of the gracefulness of M_2k, this happens, for instance, when k≡2 (mod 4) and 6kt+1 is a prime.     

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.     

Path bundles on n-cubes

A path bundle is a set of 2^a paths in an n-cube, denoted Q_n, such that every path has the same length, the paths partition the vertices of Q_n, the endpoints of the paths induce two subcubes of Q_n, and the endpoints of each path are complements. This paper shows that a path bundle exists if and only if n>0 is odd and 0?a?n-⌈log₂(n+1)⌉.     

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.     

A note on the rank parity function

We provide some further theorems on the partitions generated by the rank parity function. New Bailey pairs are established, which are of independent interest.     

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.     

Enumeration of bilaterally symmetric 3-noncrossing partitions

Schützenberger’s theorem for the ordinary RSK correspondence naturally extends to Chen et al.’s correspondence for matchings and partitions. Thus the counting of bilaterally symmetric k-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for 3-noncrossing partitions, we use a different technique to develop a Maple package for 2-dimensional vacillating lattice walk enumeration problems. The package also applies to the hesitating case. As applications, we find several interesting relations for some special bilaterally symmetric partitions.     

A group-based search for solutions of the n-queens problem

The n-queens problem is a well-known problem in mathematics, yet a full search for n-queens solutions has been tackled until now using only simple algorithms (with the exception of the Rivin-Zabih algorithm). In this article, we discuss optimizations that mainly rely on group actions on the set of n-queens solutions. Most of our arguments deal with the case of toroidal queens; at the end, the application to the regular n-queens problem is discussed, and also the Rivin-Zabih algorithm.