Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A\) be an abelian group. A labeling \(f : V(G) \to A\) induces an edge labeling \(f^* : E(G) \to A\) defined by \(f^*(xy) = f(x) + f(y)\), for each edge \(xy \in E(G)\). For \(i \in A\), let \(v_f(i) = \mathrm{card}\{v \in V(G) : f(v) = i\}\) and \(e_f(i) = \mathrm{card}\{e \in E(G) : f^*(e) = i\}\). Let \(c(f) = \{|e_f(i) – e_f(j)|: (i, j) \in A \times A\}\). A labeling \(f\) of a graph \(G\) is said to be \(A\)-friendly if \(|v_f(i)- v_f(j)| \leq 1\) for all \((i, j) \in A \times A\). If \(c(f)\) is a \((0, 1)\)-matrix for an \(A\)-friendly labeling \(f\), then \(f\) is said to be \(A\)-cordial. When \(A = \mathbb{Z}_2\), the friendly index set of the graph \(G\), \(FI(G)\), is defined as \(\{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\}\). In [13] the friendly index set of cycles are completely determined. In this paper we describe the friendly index sets of cycles with parallel chords. We show that for a cycle with an arbitrary non-empty set of parallel chords, the numbers in its friendly index set form an arithmetic progression with common difference 2.

This paper investigates the dihedral group as the array stabilizer of an augmented \(k\)-set of mutually orthogonal Latin squares. Necessary conditions for the stabilizer to be a dihedral group are established. A set of two-variable identities essential for a dihedral group to be contained in an array stabilizer are determined. Infinite classes of models that satisfy the identities are constructed.

The eccentricity \(e(v)\) of a vertex \(v\) in a connected graph \(G\) is the distance between \(v\) and a vertex furthest from \(v\). The center \(C(G)\) is the subgraph induced by those vertices whose eccentricity is the radius of \(G\), denoted \(\mathrm{rad}G\), and the periphery \(P(G)\) is the subgraph induced by those vertices with eccentricity equal to the diameter of \(G\), denoted \(\mathrm{diam}G\). The annulus \(\mathrm{Ann}(G)\) is the subgraph induced by those vertices with eccentricities strictly between the radius and diameter of \(G\). In a graph \(G\) where \(\mathrm{rad}G < \mathrm{diam}G\), the interior of \(G\) is the subgraph \(\mathrm{Int}(G)\) induced by the vertices \(v\) with \(e(v) < \mathrm{diam}G\). Otherwise, if \(\mathrm{rad}G = \mathrm{diam}G\), then \(\mathrm{Int}(G) = G\). Another subgraph for a connected graph \(G\) with \(\mathrm{rad}G < \mathrm{diam}G\), called the exterior of \(G\), is defined as the subgraph \(\mathrm{Ext}(G)\) induced by the vertices \(v\) with \(\mathrm{rad}G < e(v)\). As with the interior, if \(\mathrm{rad}G = \mathrm{diam}G\), then \(\mathrm{Ext}(G) = G\). In this paper, the annulus, interior, and exterior subgraphs in trees are characterized.

A proper vertex coloring of a graph \(G = (V, E)\) is acyclic if \(G\) contains no bicolored cycle. A graph \(G\) is acyclically \(L\)-list colorable if for a given list assignment \(L = \{L(v) : v \in V\}\), there exists a proper acyclic coloring \(\phi\) of \(G\) such that \(\phi(v) \in L(v)\) for all \(v \in V(G)\). If \(G\) is acyclically \(L\)-list colorable for any list assignment with \(|L(v)| = k\) for all \(v \in V\), then \(G\) is acyclically \(k\)-choosable. In this paper, it is proved that every toroidal graph without 4- and 6-cycles is acyclically \(5\)-choosable.

The centro-polyhedral group \(\langle l,m,n\rangle\), for \(l, m, n \in \mathbb{Z}\), is defined by the presentation
\[\langle x, y, z : x^l = y^m = z^n = xyz \rangle.\]
In this paper, we obtain the periods of \(k\)-nacci sequences in centro-polyhedral groups and related groups.

We study two-path convexity in bipartite tournaments. For a bipartite tournament, we obtain both a necessary condition and a sufficient condition on the adjacency matrix for its rank to be two. We then investigate 4-cycles in bipartite tournaments of small rank. We show that every vertex in a bipartite tournament of rank two lies on a four cycle, and bipartite tournaments with a maximum number of 4-cycles do not necessarily have minimum rank.

A set \(S\) of vertices in a graph \(G\) is a clique dominating set of \(G\) if \(S\) contains at least one vertex of every clique \(C\) of \(G\). The clique domination number \(\gamma_q(G)\) and the upper clique domination number \(\gamma_q(G)\) are, respectively, the minimum and maximum cardinalities of a minimal clique dominating set of \(G\). In this paper, we prove that the problem of computing \(\Gamma_q(G)\) is NP-complete even for split graphs and the problem of computing \(\gamma_q(G)\) is NP-complete even for chordal graphs. In addition, for a block graph \(BG\) we show that the clique domination number is bounded above by the vertex independence number (\(\gamma_q(BG) \leq \beta(BG)\)) and give a linear algorithm for computing \(\gamma_q(BG)\).

Katerinis established the following result in [1]. Let \(G\) be a simple graph with \(\delta(G) \geq \lfloor\frac{|V(G)|}{2}+k\), where \(k\) is a non-negative integer. Let \(f : V(G) \to \mathbb{Z}^+\) be a function having the following properties:

(1) \(\frac{1}{2}\left({d_G(v) – (k+1)}{2}\right) \leq f(v) \leq \frac{1}{2}\left({d_G(v) + (k+1)}{2}\right)\) for every \(v \in V(G)\),

(2) \(\sum_{v\in V(G)} f(v) = |E(G)|\).

Then \(G\) has an orientation \(D\) such that \(d^+_D(v) = f(v)\), for every \(v \in V(G)\). In this paper, we focus on the sharpness of the above two inequalities.

A cosimple regular matroid \(M\) does not have disjoint circuits if and only if \(M \in \{M(K_{3,3}), M^*(K_n) \mid n \geq 3\}\). This extends a former result of Erdős and Pósa on graphs without disjoint circuits.

Suppose \(G\) is a finite graph with vertex-set \(V(G)\) and edge-set \(E(G)\). An \((a, d)\)-edge-antimagic total labeling on \(G\) is a one-to-one map \(f\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, |V(G)| + |E(G)|\) with the property that the edge-weights \(w(uv) = f(u) + f(v) + f(uv)\), \(uv \in E(G)\), form an arithmetic progression starting from \(a\) and having common difference \(d\). Such a labeling is called super if the smallest labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labelings of disjoint union of multiple copies of complete bipartite graph.