2- and 3-existentially closed tournaments

A tournament has property $P_{k}\,  (k\geq 1)$ if for every $k$-subset $A$ of its vertices  and every $B\subseteq A$, there exists $x\notin A$ such that $x$ dominates every element of $B$ and every element of $A\setminus B$ dominates $x$. A tournament has property $S_{k}$ if $B=\varnothing$ in the definition before. We give a characterization of those circulant tournaments of prime order having property $P_{2}$ using some results of additive number theory. Some new theoretical results are proved. It is proved that in vertex-transitive doubly regular tournaments properties $S_{3}$ and $P_{3}$ are equivalent and consequently, the Paley tournament $QR_{p}$ has property $P_{3}$ for every $p\equiv 3 \bmod 4$ such that $p\geq 19$. It is also shown that the out- and in-neighborhood of every vertex of $QR_{p}$ induce a circulant tournament with a special structure. As corollaries, we obtain that the out- and in-neighborhood of every vertex of $QR_{p}$ has property $S_{3}$ if and only if $QR_{p}$ has property $S_{4}$ and that $QR_{67}$ has property $S_{4}$. In addition, non-vertex-transitive doubly regular tournaments of Szekeres type are considered. We show that the infinite families of Szekeres tournaments and their converses satisfy property $P_{3}$.

The logarithm of the exponential generating function of Eulerian polynomials

In this note we determine the series expansion of the logarithm of the exponential generating function of Eulerian polynomials, which results in a new identity on Eulerian polynomials. We also obtain similar results for general Eulerian polynomials introduced by Xiong, Tsao, and Hall. As consequences, we derive some relations between classical Eulerian polynomials and their variations

On identifying vertices of tournament digraphs

An identifying code in a graph is a subset of its vertices where the neighbours\u27 intersections with the subset are nonempty and different for every pair of vertices. After their introduction in 1998 by Karpovsky et al., the interest in this domain has never ended. This growing interest comes from, on the one hand, the theoretical aspect of this concept, and on the other hand, its applications, especially the indoor location and faulty processor network.   In this work, we study the identifying code on tournament digraphs which is probably the most studied class of digraphs. Hence, we give some minimum cardinality of special tournaments and show that only transitive tournaments can admit an $r$-identifying code when $r\geq 2$. We also obtain an upper bound for the quadratic residue tournament. Moreover, we study how to reach an optimal code when adding a vertex or inverting an arc in a transitive tournament

On the degree distance matrix of connected graphs

Let $d_u$ denote the degree of vertex $u$ and $d_{uv}$ be the distance between vertices $u$ and $v$ in a connected graph $G$. We propose studying the degree distance matrix of a connected graph $G$, defined as $M_{DD}(G) = \left((d_u + d_v)d_{uv}\right)_{u,v \in V(G)}$. This study sheds new light on the spectra of degree and distance-based matrices. Some spectral properties of $M_{DD}(G)$ are given along with some open problems that can help to understand the degree distance matrix in depth. Furthermore, $M_{DD}$ spectra of some graphs are obtained. Moreover, an effort is made to get some sharp lower and upper bounds for the $M_{DD}$ spectral radius

Uniform convergence of an asymptotic approximation to associated Stirling numbers

Let $S_r(p,q)$ be the $r$-associated Stirling numbers of the second kind, the number of ways to partition a set of size $p$ into $q$ subsets of size at least $r$. For $r=1$, these are the standard Stirling numbers of the second kind, and for $r=2$, these are also known as the Ward Numbers. This paper concerns asymptotic expansions of these Stirling numbers; such expansions have been known for many years. However, while uniform convergence of these expansions was conjectured by Hennecart, it has not been fully proved. A recent paper by Connamacher and Dobrosotskaya went a long way by proving uniform convergence on a large set. In this paper, we build on that paper and prove convergence "everywhere"

New constructions of Meyniel extremal families of graphs

We provide new constructions of Meyniel extremal graphs, which are families of graphs with the conjectured largest asymptotic cop number.  Using spanning subgraphs, we prove that there are an exponential number of new Meyniel extremal families with specified degrees. Using linear programming on hypergraphs, we explore the degrees in families that are not Meyniel extremal. We give the best-known upper bound on the cop number of vertex-transitive graphs with a prescribed degree. We find new Meyniel extremal families of regular graphs with large chromatic number, large diameter, and explore the connection between Meyniel extremal graphs and bipartite graphs. Conjectures relating Meyniel extremal families to maximum and average degrees in their graphs are presented.

Star coloring of some toroidal graphs

A proper coloring of a graph is a star coloring if there is no bicolored path on four vertices, or, equivalently, if every connected subgraph induced by any two color classes is a star. We investigate the star chromatic number $\chi_s$ of some well-known toroidal graphs. First, it is known that for the $d$-dimensional toroidal grid $TG_d$ the star chromatic number is $\mathcal{O}(d^2)$. Some results published in the literature that are applicable to this family of graphs improve this bound to $\mathcal{O}(d^{\frac{3}{2}})$. In this article we show that $\chi_s(TG_d)=\mathcal{O}(d)$. Furthermore, we investigate the star chromatic number of the honeycomb torus $HT(n)$ of size $n$, and show that $\chi_s(HT(n))=4$

Average edge order of normal 33-pseudomanifolds

In their work, Feng Luo and Richard Stong introduced the concept of the average edge order, denoted as $\mu_0$. They demonstrated that if $\mu_0(K)\leq \frac{9}{2}$ for a closed triangulated $3$-manifold $K$, then $K$ must be a sphere. Building upon this foundation, Makoto Tamura extended similar results to compact triangulated $3$-manifolds with nonempty boundaries in \cite{Tamura1, Tamura2}. In our present study, we extend these findings to normal $3$-pseudomanifolds. Specifically, we establish that for a normal $3$-pseudomanifold $K$ with singularities, $\mu_0(K)\geq\frac{30}{7}$. Moreover, equality holds if and only if $K$ is a one-vertex suspension of a triangulation of  $\mathbb{RP}^2$ with seven vertices. Furthermore, we establish that when $\frac{30}{7}\leq\mu_0(K)\leq\frac{9}{2}$, the $3$-pseudomanifold $K$ can be derived from some boundary complexes of $4$-simplices by a sequence of possible operations, including connected sums, bistellar $1$-moves, edge contractions, edge expansions, vertex folding, and edge folding

Generalized combinatorial identities for split (n+tn+t)-color partitions

This paper studies three generalized $q$-series combinatorially using split ($n+t$)-color partitions as a combinatorial tool. This work provides a generalized approach to unify the several combinatorial identities found in the literature. In this process, we obtain several new Rogers–Ramanujan–MacMahon type partition identities

A finite-bound partition identity generalizing a problem by Andrews and Deutsch

We introduce a finite-bound extension of a partition identity which was originally proposed as a problem by Andrews and Deutsch in 2016, and given a generalized form in 2018 by Smoot and Yang. We also give a simple bijective extension of the original proof