If \(G\) and \(H\) are graphs, define the Ramsey number \(r(G, H)\) to be the least number \(p\) such that if the edges of the complete graph \(K_p\) are colored red and blue (say), either the red graph contains a copy of \(G\), or the blue graph contains a copy of \(H\). In this paper, we determine the Ramsey number \(r(mC_4, nC_5)\) for any \(m\geq1, n\geq1\).

We construct a complex \(K^n\) of \(m\)-ary relations, \(1 \leq m \leq n+1\), in a finite set \(X \neq 0\), representing a model of an abstract cellular complex. For such a complex \(K^n\) we define the matrices of incidence and coincidence, the groups of homologies \(\mathcal{H}_m(K^n)\) and cohomologies \(\mathcal{H}^m(K^n)\) on the group of integers \(\mathbf{Z}\), and the Euler characteristic. On a combinatorial basis we derive their main properties. In further publications we will derive more analogues of classical properties, and also applications with respect to the existence of fixed relations in the utilization of the isomorphisms will be investigated. In particular, we intend to complete the theory of hypergraphs with the help of such topological observations.

Colbourn introduced \(V_\lambda(m, t)\) to construct transversal designs with index \(\lambda\). A \(V_\lambda(m, t)\) leads to a \((mt + 1, mt + 2; \lambda,0; t)\)-aussie-difference matrix. In this article, we use Weil’s theorem on character sums to show that for any integer \(\lambda \geq 2\), a \(V_\lambda(m, t)\) always exists in \(GF(mt + 1)\) for any prime power \(mt+1 > B_\lambda(m) = \left[\frac{E+\sqrt{E^2+4F}}{2}\right]^2\), where \(E = \lambda(u-1)(m-1)m^u-m^{u-1}+1,F=(u-1)\lambda m^u\) and \(u = \left\lfloor\frac{m{\lambda}+1+(-1)^{\lambda+1}}{2}\right\rfloor\). In particular, we determine the existence of \(V_{\lambda}(m, t)\) for \((\lambda, m) = (2, 2), (2, 3)\).

A weighted graph \((G,w)\) is a graph \(G = (V, E)\) together with a positive weight-function on its vertices \(w: V \to \mathbf{R}^{>0}\). The weighted domination number \(\gamma_w(G)\) of \((G, w)\) is the minimum weight \(w(D) = \sum_{v \in D} w(v)\) of a vertex set \(D \subseteq V\) with \(N[D] = V\), i.e. a dominating set of \(G\).

For this natural generalization of the well-known domination number, we study some of the classical questions of domination theory. We characterize all extremal graphs for the simple Ore-like bound \(\gamma_w(G) \leq \frac{1}{2}w(V)\) and prove Nordhaus-Gaddum-type inequalities for the weighted domination number.

Generalized Steiner systems GS\(_d(t,k,v,g)\) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size \(g + 1\) with minimum Hamming distance \(d\), in which each codeword has length \(v\) and weight \(k\). It was proved that the necessary conditions for the existence of a GS\(_4(2,4,v,g)\) are also sufficient for \(g = 2, 3\) and \(6\). In this paper, a general result on the existence of a GS\(_4(2,4,v,g)\) is presented. By using this result, we prove that the necessary conditions \(v \equiv 1 \pmod{3}\) and \(v \geq 7\) are also sufficient for the existence of a GS\(_4(2, 4, v, 4)\).

An \(A^3\)-code is an extension of an \(A\)-code in which none of the three participants, transmitter, receiver, and arbiter, is trusted. In this paper, we extend the previous model of \(A^3\)-codes by allowing the transmitter and the receiver to not only individually attack the system, but also collude with the arbiter against the other. We derive information-theoretic lower bounds on the success probability of various attacks, and combinatorial lower bounds on the size of key spaces. We also study the combinatorial structure of optimal \(A^3\)-codes against collusion attacks and give a construction of an optimal code.

We prove that \(15\) is the maximal size of a \(3\)-arc in the projective plane of order \(8\).

A \(k\)-line-distinguishing coloring of a graph \(G = (V,E)\) is a partition of \(V\) into \(k\) sets \(V_1, V_2, \ldots, V_k\) such that \(q(\langle V_i \rangle) \leq 1\) for \(i = 1, \ldots, k\) and \(q(V_i . V_j) \leq 1\) for \(1 \leq i < j \leq k\). If the color classes in a line-distinguishing coloring are also independent. Then it is called a harmonious coloring. A coloring is minimal if when two color classes are combined, we no longer have a coloring of the given type. The upper harmonious chromatic number, \(H(G)\), is defined as the maximum cardinality of a minimal harmonious coloring of a graph \(G\). While the upper line-distinguishing chromatic number, \(H'(G)\), is defined as the maximum cardinality of a minimal line-distinguishing coloring of a graph \(G\). We determine \(H'(C_n)\) and \(H(C_n)\) for an even cycle \(C_n\).

In this paper, we derive an inequality on the existence of bi-level balanced arrays (B-arrays) of strength eight by using a result involving central moments from statistics, and by counting in two ways the number of coincidences of various columns with a specific column. We discuss the use of this inequality in obtaining the maximum number of constraints for these arrays, and present some illustrative examples.

In this note, we present many uniquely \(n\)-colorable graphs with \(m\) vertices and new constructing ways of uniquely colorable graphs by using the theory of adjoint polynomials of graphs. We give new constructing ways of two uniquely colorable graphs which are chromatically equivalent also.

An \(LD(n,k,p,t:b)\) Lotto design is a set of \(b\) \(k\)-sets (blocks) of an \(n\)-set such that any \(p\)-set intersects at least one block in \(t\) or more elements. Let \({L}(n,k,p,t)\) denote the minimum number of blocks for any \(LD(n,k,p,t:b)\) Lotto design. This paper describes an algorithm used to construct Lotto designs by combining genetic algorithms and simulated annealing and provides some experimental results.

A union closed (UC) family \(\mathcal{A}\) is a finite family of sets such that the union of any two sets in \(\mathcal{A}\) is also in \(\mathcal{A}\). Peter Frankl conjectured that for every union closed family \(\mathcal{A}\), there exists some \(x\) contained in at least half the members of \(\mathcal{A}\). This is the union-closed sets conjecture.

An FC family is a UC family \(\mathcal{B}\) such that for every UC family \(\mathcal{A}\), if \(\mathcal{B} \subseteq \mathcal{A}\), then \(\mathcal{A}\) satisfies the union-closed sets conjecture. We give a heuristic method for identifying possible FC families, and apply it to families in \(\mathcal{P}(5)\) and \(\mathcal{P}(6)\).

The vertices \(V\) of trees with maximum degree three and \(t\) degree two vertices are partitioned into sets \(R\), \(B\), and \(U\) such that the induced subgraphs \(\langle V – R \rangle\) and \(\langle V – B \rangle\) are isomorphic and \(|U|\) is minimum. It is shown for \(t \geq 2\) that there is such a partition for which \(|U| = 0\) if \(t\) is even and \(|U| = 1\) if \(t\) is odd. This extends earlier work by the authors which answered this problem when \(t = 0\) or \(1\).

Let G be a simple connected graph on 2n vertices with a perfect matching. For a positive integer k, \(1 \leq \text{k} \leq \text{n}-1\), G is \(k\)-extendable if for every matching M of size k in G, there is a perfect matching in G containing all the edges of M. For an integer k, \(0 \leq \text{k} \leq \text{n} – 2\), G is trongly \(k\)-extendable if \(\text{G} – \{\text{u, v}\}\) is \(k\)-extendable for every pair of vertices u and v of G. The problem that arises is that of characterizing k-extendable graphs and strongly k-extendable graphs. The first of these problems has been considered by several authors while the latter has been recently investigated. In this paper, we focus on a minimum cutset of strongly k-extendable graphs. For a minimum cutset S of a strongly k-extendable graph G, we establish that if \(|\text{S}| = \text{k + t}\), for an integer \(\text{t} \geq 3\), then the independence number of the induced subgraph G[S] is at most \(2\) or at least k + 5 – t. Further, we present an upper bound on the number of components of G – S.

Let \(\{G_{pn} | n \geq 1\} = \{G_{p1}, G_{p2}, G_{p3}, \ldots\}\) be a countable sequence of simple graphs, where \(G_{pn}\) has \(pn\) vertices. This sequence is called \(K_p\)-removable if \(G_{p1} = K_p\), and \(G_{pn} – K_p = G_{p(n-1)}\) for every \(n \geq 2\) and for every \(K_p\) in \(G_{pn}\). We give a general construction of such sequences. We specialize to sequences in which each \(G_{pn}\) is regular; these are called regular \((K_p, \lambda)\)-removable sequences, where \(\lambda$ is a fixed number, \(0 \leq \lambda \leq p\), referring to the fact that \(G_{pn}\) is \((\lambda(n – 1) + p – 1)\)-regular. We classify regular \((K_p, 0)\)-, \((K_p, p – 1)\)-, and \((K_p, p)\)-removable sequences as the sequences \(\{nK_p | n \geq 1\}\), \(\{K_{p \times n} | n \geq 1\}\), and \(\{K_{pn} | n \geq 1\}\) respectively. Regular sequences are also constructed using `levelled’ Cayley graphs, based on a finite group. Some examples are given.

A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(u, v,\) and \(w\) with \(d(u,v) = 2\) and \(w \in N(u) \cap N(v)\), \(d(u) + d(v) \geq |N(u) \cup N(v) \cup N(w)| – 1\). Let \(G\) be a 2-connected \(L_1\)-graph of order \(n\). If \(\sigma_3(G) \geq n – 2\), then \(G\) is hamiltonian or \(G \in \mathcal{K}\), where \(\sigma_3(G) = \min\{d(u) + d(v) + d(w) : \{u,v,w\} \text{ is an independent set in } G\}\), \(\mathcal{K}=\{G: K_{p, p+1} \subseteq G \subseteq K_p + (p+1)K_1 for some p \geq 2\}\). A similar result on the traceability of connected \(L_1\)-graphs is also obtained.

For a graph \(G\), the jump graph \(J(G)\) is that graph whose vertices are the edges of \(G\) and where two vertices of \(J(G)\) are adjacent if the corresponding edges are not adjacent. For \(k \geq 2\), the \(k\)th iterated jump graph \(J^k(G)\) is defined as \(J(J^{k-1}(G))\), where \(J^1(G) = J(G)\). An infinite sequence \(\{G_i\}\) of graphs is planar if every graph \(G_i\) is planar; while the sequence \(\{G_i\}\) is nonplanar otherwise. All connected graphs \(G\) for which \(\{J^k(G)\}\) is planar have been determined. In this paper, we investigate those connected graphs \(G\) for which \(\{J^k(G)\}\) is nonplanar. It is shown that if \(\{J^k(G)\}\) is a nonplanar sequence, then \(J^k(G)\) is nonplanar for all \(k \geq 4\). Furthermore, there are only six connected graphs \(G\) for which \(\{J^k(G)\}\) is nonplanar and \(J^3(G)\) is planar.

We examine a query posed as a conjecture by Key and Moori [11, Section 7] concerning the full automorphism groups of designs and codes arising from primitive permutation representations of finite simple groups, and based on results for the Janko groups \(J_1\) and \(J_2\) as studied in [11]. Here, following that same method of construction, we show that counter-examples to the conjecture exist amongst some representations of some alternating groups, and that the simple symplectic groups in their natural representation provide an infinite class of counter-examples.