Ryser’s Conjecture states that for any \( r \)-partite \( r \)-uniform hypergraph, the vertex cover number is at most \( r-1 \) times the matching number. This conjecture is only known to be true for \( r \leq 3 \). For intersecting hypergraphs, Ryser’s Conjecture reduces to saying that the edges of every \( r \)-partite intersecting hypergraph can be covered by \( r-1 \) vertices. This special case of the conjecture has only been proven for \( r \leq 5 \).

It is interesting to study hypergraphs which are extremal in Ryser’s Conjecture, i.e., those hypergraphs for which the vertex cover number is exactly \( r-1 \) times the matching number. There are very few known constructions of such graphs. For large \( r \), the only known constructions come from projective planes and exist only when \( r-1 \) is a prime power. Mansour, Song, and Yuster studied how few edges a hypergraph which is extremal for Ryser’s Conjecture can have. They defined \( f(r) \) as the minimum integer so that there exists an \( r \)-partite intersecting hypergraph \( \mathcal{H} \) with \( \tau(\mathcal{H}) = r-1 \) and with \( f(r) \) edges. They showed that \( f(3) = 3 \), \( f(4) = 6 \), \( f(5) = 9 \), and \( 12 \leq f(6) \leq 15 \).

In this paper, we focus on the cases when \( r = 6, 7 \), and \( 11 \). We show that \( f(6) = 13 \), improving previous bounds. Also, by providing the first known extremal hypergraphs for the \( r = 7 \) and \( r = 11 \) case of Ryser’s Conjecture, we show that \( f(7) \leq 22 \) and \( f(11) \leq 51 \). Our results for \( f(6) \) and \( f(7) \) have been obtained independently by Aharoni, Barat, and Wanless.

Let \( T = S \setminus \left( \cup \left\{ A : A \, \text{ in } \, \mathcal{A} \right\} \right) \), where \( S \) is an orthogonal polytope in \( \mathbb{R}^d \) for \( d \geq 2 \) and where \( \mathcal{A} \) is a collection of \( n \) pairwise disjoint open boxes contained in \( S \). Point \( x \) belongs to \(\text{Ker } T \) if and only if \( x \) belongs to \(\text{Ker } S \) and no coordinate line at \( x \) meets any \( A \) in \( \mathcal{A} \). In turn, this relationship between the staircase kernels of \( S \) and \( T \) produces a Krasnosel’skii-type result for \( T \) in terms of \( n \), extending the class of orthogonal polytopes for which such a theorem exists.

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to \(K_{1,3}\). Let \(k\) be an integer with \(k \geq 2\). We prove that if \(G\) is a claw-free graph of order at least \(14k – 13\) and with minimum degree at least four, then \(G\) contains \(k\) vertex-disjoint copies of \(K_{1,4}\). This partially supports a conjecture proposed by Jiang, Chiba, Fujita and Yan.

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.