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.