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.