We show that the classical Ramsey number \(R(3,3,3,3)\) is no greater than \(62\). That is, any edge coloring with four colors of a complete graph on \(62\) vertices must contain a monochromatic triangle. Basic notions and a historical overview are given along with the theoretical framework underlying the main result. The algorithms for the computational verification of the result are presented along with a brief discussion of the software tools that were utilized.

We show that certain subsets of \(\mathbf{F}_q\)-rational points of the curve \(XZ^{n-1} = Y^n\) are dense sets in \(\mathbf{P}^2(\mathbf{F}_q)\).

H. Kharaghani, in “Arrays for orthogonal designs” \((2000)\), \(\textit{J. Combin. Designs}\), \(8 (2000), 166-173\), showed how to use amicable sets of matrices to construct orthogonal designs in orders divisible by eight. We show how amicable orthogonal designs can be used to make amicable sets and so obtain infinite families of orthogonal designs in six variables in orders divisible by eight.

Let \(G\) and \(H\) be a pair of non-isomorphic graphs on fewer than \(m\) vertices. In this paper, we introduce several new problems about decomposing the complete graph \(K_m\) into copies of \(G\) and \(H\). We will assume that at least one of \(G\) or \(H\) is not a cycle. We also begin to examine variations to the problems of subgraph packing, covering, and factorization.

For vertices \(u\) and \(v\) in a connected graph \(G\) with vertex set \(V\), the distance \(d(u,v)\) is the length of a shortest \(u – v\) path in \(G\). A \(u – v\) path of length \(d(u,v)\) is called a \(u – v\) geodesic. The closed interval \(I[u,v]\) consists of \(u\), \(v\), and all vertices that lie in some \(u – v\) geodesic of \(G\); while for \(S \subseteq V\), \(I[S]\) is the union of closed intervals \(I[u,v]\) for all \(u,v \in S\). A set \(S\) of vertices is a geodetic set if \(I[S] = V\), and the minimum cardinality of a geodetic set is the geodetic number \(g(G)\). For vertices \(x\) and \(y\) in \(G\), the detour distance \(D(x, y)\) is the length of a longest \(x – y\) path in \(G\). An \(x – y\) path of length \(D(x, y)\) is called an \(x – y\) detour. The closed detour interval \(I_D[x,y]\) consists of \(x\), \(y\), and all vertices in some \(x – y\) detour of \(G\). For \(S \subseteq V\), \(I_D[S]\) is the union of \(I_D[x,y]\) for all \(x,y \in S\). A set \(S\) of vertices is a detour set if \(I_D[S] = V\), and the minimum cardinality of a detour set is the detour number \(dn(G)\). We study relationships that can exist between minimum detour sets and minimum geodetic sets in a graph. A graph \(F\) is a minimum detour subgraph if there exists a graph \(G\) containing \(F\) as an induced subgraph such that \(V(F)\) is a minimum detour set in \(G\). It is shown that \(K_3\) and \(P_3\) are minimum detour subgraphs. It is also shown that for every pair \(a,b \geq 2\) of integers, there exists a connected graph \(G\) with \(dn(G) = a\) and \(g(G) = b\).

For a vertex \(v\) of a connected graph \(G\) and a subset \(S\) of \(V(G)\), the distance between \(v\) and \(S\) is \(d(v, S) = \min\{d(v,x) : x \in S\}\), where \(d(v,x)\) is the distance between \(v\) and \(x\). For an ordered \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\), the code of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(c_\Pi(v) = (d(v,S_1), d(v,S_2), \ldots, d(v, S_k))\). The \(k\)-partition \(\Pi\) is a resolving partition if the codes \(c_\Pi(v)\), \(v \in V(G)\), are distinct. A resolving partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) is acyclic if each subgraph \(\langle S_i \rangle\) induced by \(S_i\) (\(1 \leq i \leq k\)) is acyclic in \(G\). The minimum \(k\) for which there is a resolving acyclic \(k\)-partition of \(V(G)\) is the resolving acyclic number \(a_r(G)\) of \(G\). We study connected graphs with prescribed order, diameter, vertex-arboricity, and resolving acyclic number. It is shown that, for each triple \(d,k,n\) of integers with \(2 \leq d \leq n-2\) and \(3 \leq (n-d+1)/2 \leq k \leq n-d+1\), there exists a connected graph of order \(n\) having diameter \(d\) and resolving acyclic number \(k\). Also, for each pair \(a, b\) of integers with \(2 \leq a \leq b-1\), there exists a connected graph with resolving acyclic number \(a\) and vertex-arboricity \(b\). We present a sharp lower bound for the resolving acyclic number of a connected graph in terms of its clique number. The resolving acyclic number of the Cartesian product \(H \times K_2\) of nontrivial connected graph \(H\) and \(K_2\) is studied.

In this paper, we completely solve the problem of finding a maximum packing of any balanced complete multipartite graph \(K_{m}(n)\) with edge-disjoint \(6\)-cycles, and minimum leaves are explicitly given.

Subsequently, we also find a minimum covering of \(K_{m}(n)\).

Orthogonal designs and their special cases, such as weighing matrices and Hadamard matrices, have many applications in combinatorics, statistics, and coding theory, as well as in signal processing. In this paper, we generalize the definition of orthogonal designs, give many constructions for these designs, and prove some multiplication theorems that, most of them, can also be applied in the special case of orthogonal designs. Some necessary conditions for the existence of generalized orthogonal designs are also given.

We prove that the corona graphs \(C_n \circ K_1\) are \(k\)-equitable, as per Cahit’s definition of \(k\)-equitability, for \(k = 2, 3, 4, 5, 6\).

For a vertex \(v\) of a graph \(G = (V, E)\), the domination number \(\gamma(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a dominating set in \(G\) that contains \(v\). The average domination number of \(G\) is \(\gamma_{av}(G) = \frac{1}{|V|} \sum_{v\in V} \gamma_v(G)\). The independent domination number \(i_v(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a maximal independent set in \(G\) that contains \(v\). The average independent domination number of \(G\) is \(\gamma_{av}^i(G) = \frac{1}{|V|} \sum_{v\in V} i_v(G)\). In this paper, we show that a tree \(T\) satisfies \(\gamma_{av}(T) = i_{av}(T)\) if and only if \(A(T) = \vartheta\) or each vertex of \(A(T)\) has degree \(2\) in \(T\), where \(A(T)\) is the set of vertices of \(T\) that are contained in all its minimum dominating sets.