Two combinatorial identities are proved:
(1) \(\quad H_n(\varepsilon) = \frac{n+2}{3} M_n(\varepsilon)\), where \(H_n(\varepsilon)\) denotes the total number of vertices in all the n-edged rooted planar Eulerian maps and \(M_n(\varepsilon)\) denotes the number of such maps.
(2) \(\quad H_n(\mathcal{L}) = \frac{5n^2+13n+2}{2(4n+1)} M_{n }(\mathcal{L})\), where \(H_n(\mathcal{L})\) and \(M_{n}(\mathcal{L})\) are defined similarly for the class \(\mathcal{L}\) of loopless maps.
Simple closed formulae for \(M_n(\varepsilon)\) and \(M_{n}(\mathcal{L})\) are well known, and they correspond to equivalent binomial sum identities.

We derive the exact joint distribution and prove the asymptotic joint normality of the numbers of peaks, double rises, troughs, and double falls in a random permutation. A Chi-square randomness test, as a by-product, is also proposed.

For a graph \(G\) with vertex set \(V\), the total redundance, \(\text{TR}(G)\), and efficiency, \(\text{F}(G)\), are defined by the two expressions:
\(\text{TR}(G) = \min \left\{ \sum_{v \in S} (1 + \deg v) :S\subseteq V \text{and} [N(x) \cap S] \geq 1 \quad \forall x \in V \right\},\)
\(\text{F}(G) = \max \left\{ \sum_{v \in S} (1 + \deg v) :S\subseteq V \text{and} [N(x) \cap S] \leq 1 \quad \forall x \in V \right\}.\)
That is, \(\text{TR}\) measures the minimum possible amount of domination if every vertex is dominated at least once, and \(\text{F}\) measures the maximum number of vertices that can be dominated if no vertex is dominated more than once.

We establish sharp upper and lower bounds on \(\text{TR}(G)\) and \(\text{F}(G)\) for general graphs \(G\) and, in particular, for trees, and briefly consider related Nordhaus-Gaddum-type results.

In this paper, generalizations of edge-cordial labelings are introduced and studied for special classes of trees and graphs.

The total of \(4079\) \(2\)-designs and two \(3\)-designs on \(21\) points have been found. All these designs have the same group as an automorphism group. This group can be represented as the wreath product of \(G\) and \(H\), where \(G\) denotes the subgroup of order 3 of \(\text{PSL}(2,2)\) and \(H\) denotes the transitive subgroup of order 21 of \(\text{PSL}(3, 2)\).

In particular, \(1, 20, 101, 93, 173, 824\) and \(2867\) values of \(A\) for \(2\)-\((21,k,\lambda)\) designs have been detected, where \(k\) takes values from \(4\) through \(10\). Up to our knowledge, \(2217\) of these \(\lambda\)-values are new (\(14, 76, 65, 122, 587\), and \(1353\), for \(k\) equal to \(5, 6, …,10\), respectively). By Alltop’s extension [4], \(1353\) new \(2\)-\((21,10,A)\) designs can be extended to the same number of new \(3\)-\((22,11,\lambda)\) designs.

An extensive search with \(t > 2\) and \(k < 8\) has given only the \(3\)-\((21,6,216)\) design and the \(3\)-\((21,7,1260)\) design with the same automorphism group.

We give new sets of sequences with zero autocorrelation function and entries from the set \(\{0, \pm a, \pm b, \pm c\}\) where \(a, b\) and \(c\) are commuting variables (which may also be set zero). Then we use these sequences to construct some new orthogonal designs.

We show the known necessary conditions for the existence of an OD\((28; s_1, s_2, s_3)\) plus the condition that \((s_1, s_2, s_3) \neq (1, 5, 20)\) are sufficient conditions for the existence of an OD\((28; s_1, s_2, s_3)\). We also show the known necessary conditions for the existence of an OD\((28; s_1, s_2, s_3)\) constructed using four circulant matrices are sufficient conditions for the existence of 4-NPAF\((s_1, s_2, s_3)\) of length 2 for all lengths \(n \geq 7\).

We establish asymptotic existence results for OD\((4N; s_1, s_2)\) for \(3 \leq s_1 + s_2 \leq 28\). This leaves no cases undecided for \(1 \leq s_1 + s_2 \leq 28\). We show the known necessary conditions for the existence of an OD\((28; s_1, s_2)\) with \(25 \leq s_1 + s_2 \leq 28\), constructed using four circulant matrices, plus the condition that \((s_1, s_2) \neq (1, 26), (2, 25), (7, 19), (8, 19)\) or \((13, 14)\), are sufficient conditions for the existence of 4-NPAF\((s_1, s_2)\) of length \(n\) for all lengths \(n \geq 7\).

Let \(G = G(V, E)\) be a graph. A labeling of \(G\) is a function \(f: V \to \{0, 1, \ldots, n\}\) such that for each edge \(e = (u, v) \in E\), \(f(e) = |f(u) – f(v)|\). Such a labeling is said to be \(k\)-equitable if it is a labeling of the vertices with the numbers \(0\) through \(k – 1\) such that, if \(v_i\) is the number of vertices labeled \(i\), and \(e^i\) is the number of edges labeled \(i\), then \(|v^i – v^j| < 1\) and \(|e^i – e^j| \leq 1\) for all \(i, j\). A graph is said to be \(k\)-equitable if it has a \(k\)-equitable labeling. In this paper, we characterize the \(k\)-equitability of complete bipartite graphs and consider the equitability of complete multipartite graphs.

Let \(D\) be a connected symmetric digraph, \(A\) a finite abelian group with some specified property and \(g \in A\). We present a characterization for two \(g\)-cyclic \(A\)-covers of \(D\) to be isomorphic with respect to a group \(\Gamma\) of automorphisms of \(D\), for any \(g\) of odd order. Furthermore, we consider the number of \(\Gamma\)-isomorphism classes of \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order. We enumerate the number of isomorphism classes of \(g\)-cyclic \({Z}_{p^n}\)-covers of \(D\) with respect to the trivial group of automorphisms of \(D\), for any prime \(p (> 2)\), where \(\mathbb{Z}_{p^n}\) is the cyclic group of order \(p^n\). Finally, we count \(\Gamma\)-isomorphism classes of cyclic \({F}_p\)-covers of \(D\).

We completely settle the existence problem for group divisible designs with first and second associates in which the block size is \(3\), and with \(m\) groups each of size \(n\), where \(n, m \geq 3\).

We give a new and simple proof for the cyclic group of line crossings on the \(2-D\) torus.