A graph \( G \) is said to be \( k \)-\(\gamma\)-edge critical if the domination number \(\gamma(G) = k\) and \(\gamma(G + uv) < k\) for every \( uv \notin E(G) \). For the connected domination number \(\gamma_c(G) = k\), the total domination number \(\gamma_t(G) = k\) and the independent domination number \( i(G) = k \), a \( k \)-\(\gamma_c\)-edge critical graph, a \( k \)-\(\gamma_t\)-edge critical graph and a \( k \)-\(i\)-edge critical graph are similarly defined. In our previous work, we proved that every \( 2 \)-connected \( k \)-\(\gamma_c\)-edge critical graph is hamiltonian for \( 1 \leq k \leq 3 \) and we provided a class of \( l \)-connected \( k \)-\(\gamma_c\)-edge critical non-hamiltonian graphs for \( k \geq 4 \) and \( 2 \leq l \leq \frac{n-3}{k-1} \). The problem of interest is to determine a sufficient condition for \( k \)-\(\gamma_c\)-edge critical graphs to be hamiltonian for \( k \geq 4 \). In this paper, we prove that every \( 2 \)-connected \( 4 \)-\(\gamma_c\)-edge critical claw-free graph is hamiltonian. For \( k \geq 5 \), we provide a class of \( k \)-\(\gamma_c\)-edge critical claw-free non-hamiltonian graphs of connectivity two. We further show that all \( 3 \)-connected \( k \)-\(\gamma_c\)-edge critical claw-free graphs are hamiltonian for \( 1 \leq k \leq 6 \). Our methodology also establishes some results on the hamiltonian properties of \( 3 \)-connected \( k \)-\(\mathcal{D} \)-edge critical claw-free graphs where \( \mathcal{D} \in \{ \gamma, \gamma_t, i \} \).

A star forest is a forest each of whose components is a star. The star arboricity of a graph \(G\), denoted by \(\textrm{st}(G)\), is the minimum number of star forests whose union covers all the edges of \(G\). A nonzero element of a commutative ring \(R\) with unity is said to be a \({zero-divisor}\) of \(R\) if there exists a nonzero element \(y \in R\) such that \(xy = 0\). Given a ring \(R\) with unity, the \({zero-divisor\; graph}\) of \(R\), denoted by \(\Gamma(R)\), is the graph whose vertex set consists of the zero divisors of \(R\) and two vertices \(x, y \in V(\Gamma(R))\) are adjacent if and only if \(xy = 0\) in \(R\). This paper investigates the star arboricities of the zero divisor graphs \(\Gamma(\mathbb{Z}_{p^n})\), where \(n, p \in \mathbb{N}\) and \(p\) is a prime. In particular, we give bounds for \(\textrm{st}(\Gamma(\mathbb{Z}_{p^n}))\) when \(n\) is odd and determine the values of \(\textrm{st}(\Gamma(\mathbb{Z}_{p^n}))\) when \(n\) is even.

An adjacent vertex distinguishing total coloring of a graph \(G\) is a proper total \(k\)-coloring of \(G\) such that any two adjacent vertices have different color sets, where the color set of a vertex \(v\) contains the color of \(v\) and the colors of its incident edges. Let \(\chi_{a}^{”}(G)\) denote the smallest value \(k\) in such a coloring of \(G\). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if a planar graph \(G\) with maximum degree \(\Delta \geq 9\) contains no \(5\)-cycles with more than one chord, then \(\chi_{a}^{”}(G) \leq \Delta + 3\).

The concept of the skew energy of a digraph was introduced by Adiga, Balakrishnan and \(S_0\) in \(2010\). Let \(\overrightarrow{G}\) be an oriented graph of order \(n\) and \(\lambda_1, \lambda_2, \dots, \lambda_n\) denote all the eigenvalues of the skew-adjacency matrix of \(\overrightarrow{G}\). The skew energy \(\varepsilon_s(\overrightarrow{G}) = \sum\limits_{i=1}^{n} |\lambda_i|\). Hou, Shen and Zhang determined the minimal and the second minimal skew energy of the oriented unicyclic graphs. In this paper, the oriented unicyclic graphs with the third, fourth and fifth minimal skew energy are characterized, respectively.

For a finite group \( G \), a bijection \( \theta: G \to G \) is a \({strong \;complete \;mapping}\) if the mappings \( g \mapsto g\theta(g) \) and \( g \mapsto g^{-1}\theta(g) \) are both bijections. A group is \({strongly \;admissible}\) if it admits strong complete mappings. Strong complete mappings have several combinatorial applications. There exists a Latin square orthogonal to both the multiplication table of a finite group \( G \) and its normal multiplication table if and only if \( G \) is strongly admissible. The problem of characterizing strongly admissible groups is far from settled. In this paper, we will update progress towards its resolution. In particular, we will present several infinite classes of strongly admissible dihedral and quaternion groups and determine all strongly admissible groups of order at most \(31\).

The complete directed graph of order \(n\), denoted \({K}_n^*\), is the directed graph on \(n\) vertices that contains the arcs \((u,v)\) and \((v,u)\) for every pair of distinct vertices \(u\) and \(v\). For a given directed graph \(D\), the set of all \(n\) for which \({K}_n^*\) admits a \(D\)-decomposition is called the spectrum of \(D\). In this paper, we find the spectrum for each bipartite subgraph of \({K}_4^*\) with 5 or fewer arcs.

A bipartite graph on \(n\) vertices, with \(n\) even, is called uniquely bi-pancyclic (UBPC) if it contains precisely one cycle of length \(2m\) for every \(2 \leq m \leq \frac{n}{2}\). In this note, using computer programs, we show that if \(32 \leq n \leq 56\), and \(n \neq 44\), then there are no UBPC graphs of order \(n\). We also present the six non-isomorphic UBPC graphs of order 44. This improves the recent results on UBPC graphs of order at most 30.

A graph \(G\) with vertex set \(V\) and edge set \(E\) is called super edge-graceful if there is a bijection \(f\) from \(E\) to \(\{0, \pm 1, \pm 2, \dots, \pm (|E| – 1)/2\}\) when \(|E|\) is odd and from \(E\) to \(\{\pm1, \pm2, \dots, \pm|E|/2\}\) when \(|E|\) is even, such that the induced vertex labeling \(f^*\) defined by \(f^*(u) = \sum f(uv)\) over all edges \(uv\) is a bijection from \(V\) to \(\{0, \pm 1, \pm 2, \dots, \pm (|V| – 1)/2\}\) when \(|V|\) is odd and from \(V\) to \(\{\pm1, \pm2, \dots, \pm|V|/2\}\) when \(|V|\) is even. A kite is a graph formed by merging a cycle and a path at an endpoint of the path. In this paper, we prove that all kites with \(n \geq 5\) vertices, \(n \neq 6\), are super edge-graceful.

A graph with \(v\) vertices is \((r)\)-pancyclic if it contains precisely \(r\) cycles of every length from \(3\) to \(v\). A bipartite graph with an even number of vertices \(v\) is said to be \((r)\)-bipancyclic if it contains precisely \(r\) cycles of each even length from \(4\) to \(v\). A bipartite graph with an odd number of vertices \(v\) and minimum degree at least \(2\) is said to be oddly \((r)\)-bipancyclic if it contains precisely \(r\) cycles of each even length from \(4\) to \(v-1\). In this paper, using a computer search, we classify all \((r)\)-pancyclic and \((r)\)-bipancyclic graphs, \(r \geq 2\), with \(v\) vertices and at most \(v+5\) edges. We also classify all oddly \((r)\)-bipancyclic graphs, \(r \geq 1\), with \(v\) vertices and at most \(v+4\) edges.

A handicap distance antimagic labeling of a graph \(G = (V,E)\) with \(n\) vertices is a bijection \(f : V \to \{1,2,\dots,n\}\) with the property that \(f(x_i) = i\) and the sequence of the weights \(w(x_1), w(x_2), \dots, w(x_n)\) (where \(w(x_i) = \sum_{x_j \in N(x_i)}{f(x_j)}\)) forms an increasing arithmetic progression. A graph \(G\) is a handicap distance antimagic graph if it allows a handicap distance antimagic labeling.

We construct regular handicap distance antimagic graphs for every feasible odd order.