The covering number for a subset of leaves in a finite rooted tree is defined as the number of subtrees which remain after deleting all the paths connecting the root and the other leaves. We find the formula for the total sum (hence the average) of the covering numbers for a given subset of labeled leaves over all unordered binary trees with \( n \) leaves.

Let \( G = (V, E) \) be a connected graph. A dominating set \( S \) of \( G \) is called a \({neighborhood \;connected\; dominating\; set}\) (\({ncd-set}\)) if the induced subgraph \( \langle N(S) \rangle \) is connected, where \( N(S) \) is the open neighborhood of \( S \). A partition \( \{V_1, V_2, \ldots, V_k\} \) of \( V(G) \), in which each \( V_i \) is an ncd-set in \( G \), is called a \({neighborhood\; connected\; domatic\; partition}\) or simply \({nc-domatic \;partition}\) of \( G \). The maximum order of an nc-domatic partition of \( G \) is called the neighborhood connected domatic number (nc-domatic number) of \( G \) and is denoted by \( d_{nc}(G) \). In this paper, we initiate a study of this parameter.

In this note, we exhibit shortest single axioms for SQS-skeins and Mendelsohn ternary quasigroups that were found with the aid of the automated theorem-prover Prover9 and the finite model-finder

An injective map from the vertex set of a graph \( G \) to the set of all natural numbers is called an arithmetic/geometric labeling of \( G \) if the set of all numbers, each of which is the sum or product of the integers assigned to the ends of some edge, form an arithmetic/geometric progression. A graph is called arithmetic/geometric if it admits an arithmetic/geometric labeling. In this note, we show that the two notions just mentioned are equivalent—i.e., a graph is arithmetic if and only if it is geometric.

For given graphs \( G_1 \) and \( G_2 \), the \( 2 \)-color Ramsey number \( R(G_1, G_2) \) is defined to be the least positive integer \( n \) such that every \( 2 \)-coloring of the edges of the complete graph \( K_n \) contains a copy of \( G_1 \) colored with the first color or a copy of \( G_2 \) colored with the second color. In this note, we obtained some new exact values of generalized Ramsey numbers such as cycle versus book, book versus book, and complete bipartite graph versus complete bipartite graph.

We show that the necessary conditions are sufficient for the existence of group divisible designs (PBIBDs of group divisible type) for block size \( k = 3 \) and with three groups of sizes \( 1 \), \( 1 \), and \( n \).

Let \( \mathcal{B} \subseteq 2^{[m]} \) be an antichain of size \( |\mathcal{B}| =: n \). \( 2^{[m]} \) is ordered by inclusion. An antichain \( \mathcal{B} \) is called \( k \)-regular (\( k \in \mathbb{N} \)), if for each \( i \in [m] \) there are exactly \( k \) sets \( B_1, B_2, \ldots, B_k \in \mathcal{B} \) containing \( i \). In this case, we say that \( \mathcal{B} \) is a \( (k, m, n) \)-antichain.

Let \( m \geq 2 \) be an arbitrary natural number. In this note, we show that an \( (m-1, m, n) \)-antichain exists if and only if \( n \in [m+2, \binom{m}{2} – 2] \cup \{m, \binom{m}{2}\} \).

Let \( G = (V, E) \) be a connected graph. A subset \( S \) of \( V \) is called a degree equitable set if the degrees of any two vertices in \( S \) differ by at most one. The minimum order of a partition of \( V \) into independent degree equitable sets is called the \({degree \;equitable\; chromatic \;number}\) of \( G \) and is denoted by \( \chi_{de}(G) \). In this paper, we initiate a study of this new coloring parameter.

An avoidance problem of configurations in \( 4 \)-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erdős’ \( r \)-sparse conjecture on Steiner triple systems. A \( 4 \)-cycle system of order \( v \), \( 4CS(v) \), is said to be \( r \)-sparse if for every integer \( j \) satisfying \( 2 \leq j \leq r \) it contains no configurations consisting of \( j \) \( 4 \)-cycles whose union contains precisely \( j + 3 \) vertices. If an \( r \)-sparse \( 4CS(v) \) is also free from copies of a configuration on two \( 4 \)-cycles sharing a diagonal, called the double-diamond, we say it is strictly \( r \)-sparse. In this paper, we show that for every admissible order \( v \) there exists a strictly \( 4 \)-sparse \( 4CS(v) \). We also prove that for any positive integer \( r \geq 2 \) and sufficiently large integer \( v \), there exists a constant number \( c \) such that there exists a strictly \( r \)-sparse \( 4 \)-cycle packing of order \( v \) with \( c \cdot v^2 \) \( 4 \)-cycles.

A set of Hamilton cycles in the complete graph \( K_n \) is called a Dudeney set if every path of length two lies on exactly one of the cycles. It has been conjectured that there is a Dudeney set for every complete graph. It is known that there exists a Dudeney set for \( K_n \) when \( n \) is even, but the question is still unsettled when \( n \) is odd.

In this paper, we define a black \( 1 \)-factor in \( K_{p+1} \) for an odd prime \( p \), and show that if there exists a black \( 1 \)-factor in \( K_{p+1} \), then we can construct a Dudeney set for \( K_{p+2} \). We also show that if there is a black \( 1 \)-factor in \( K_{p+1} \), then \( 2 \) is a quadratic residue modulo \( p \). Using this result, we obtain some new Dudeney sets for \( K_n \) when \( n \) is odd.