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.
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.
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.
An identifying code in a graph is a subset of its vertices where the neighbours' 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 >= 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.
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 MDD(G) = ((d(u) + d(v))d(uv))(u,v is an element of V (G)). This study sheds new light on the spectra of degree and distance-based matrices. Some spectral properties of MDD(G) are given along with some open problems that can help to understand the degree distance matrix in depth. Furthermore, MDD 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.
A tournament has property P-k (k >= 1) if for every k-subset A of its vertices and every B subset of A, there exists x is not an element of A such that x dominates every element of B and every element of A \ B dominates x. A tournament has property S-k if B = & empty; 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 equivalent to 3 mod 4 such that p >= 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 S4 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.
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 O(d(2)). Some results published in the literature that are applicable to this family of graphs improve this bound to O(d(3/2)). In this article we show that chi(s)(TG(d)) = 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.
This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are minimal prime: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete description of such classes. In fact, each one of these classes is a well-quasi-ordered age and there are uncountably many of them. Eleven of these ages are almost multichainable; they remain well-quasi-ordered when labels from a well-quasi-ordering are added, hence have finitely many bounds. Five ages among them are exhaustible. Among the remaining ones, only countably many remain well-quasi-ordered when one label is added, and these have finitely many bounds (except for the age of the infinite path and its complement). The others have infinitely many bounds. Except for six examples, members of these ages we characterize are permutation graphs. In fact, every age which is not among the eleven ones is the age of a graph associated to a uniformly recurrent word on the integers. A description of minimal prime classes of posets and bichains is also provided. Our results hint towards the truth of three conjectures. One stating that if a hereditary class of finite graphs does not remain well-quasi-ordered when adding labels in a well-quasi ordered set to these graphs, then it is not well-quasi-ordered when adding just two constants to each of these graphs.
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.
We introduce a variation of the Cops and Robber game in which the robber side consists of a robber and a decoy which are indistinguishable to the cops except under certain conditions. The cops win when one of them moves onto the same vertex as the actual robber (i.e. not the decoy) after a finite number of turns. The robber can throw the decoy to a neighbouring vertex on any turn beyond his first; such a turn for the robber consists of throwing (or dropping) the decoy and then moving. The current decoy disappears as the next is thrown so there is only a single decoy in play at any time. We characterize decoycopwin graphs in the case where the cop can distinguish between the robber and decoy only when he is on the same vertex as one of them. We also characterize such graphs if the cop can distinguish between the robber and decoy only when he has cornered at least one of them.
In this article, we study certain matrices whose entries are Stirling numbers of the second kind; these are called Stirling-like matrices of the second kind. We obtain, among other results, several matrix decompositions of these matrices and evaluate their determinants. In particular, we find some identities for 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".
In their work, Feng Luo and Richard Stong introduced the concept of the average edge order, denoted as & micro;(0). They demonstrated that if & micro;(0)(K) <= 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 [12, 13]. In our present study, we extend these findings to normal 3-pseudomanifolds. Specifically, we establish that for a normal 3-pseudomanifold K with singularities, & micro;(0)(K) >= 30/7. Moreover, equality holds if and only if K is a one-vertex suspension of a triangulation of RP2 with seven vertices. Furthermore, we establish that when 30/7 <= & micro;(0)(K) <= 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.
We give conditions for a locally finite poset $P$ to have the property that for any functions $f:P\to {\bf C}$ and $g:P\to {\bf C}$ not identically zero and linked by the M\"obius inversion formula, the support of at least one of $f$ and $g$ is infinite. This generalises and gives an entirely poset-theoretic proof of a result of Pollack. Various examples and non-examples are discussed.
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 a linear programming problem 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.
We calculate the Hankel determinants of sequences of Bernoulli polynomials. This corresponding Hankel matrix comes from statistically estimating the variance in nonparametric regression. Besides its entries’ natural and deep connection with Bernoulli polynomials, a special case of the matrix can be constructed from a corresponding Vandermonde matrix. As a result, instead of asymptotic analysis, we give a direct proof of calculating an entry of its inverse. Further extensions also include an identity of Stirling numbers of the both kinds.
In this paper, we present an algorithm that allows us to compute the permanent of a tensor by using Laplace expansion. We prove that the permanent of a $4$-dimensional polystochastic $(0,1)$-tensor of order $n$ constructed using a special $n\times (n-1)$ row-Latin rectangle $R$ with no transversals is positive. Also, we show that the permanent of an even-dimensional polystochastic $(0,1)$-tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. The result obtained here proves that each odd-dimensional Latin hypercube of order $4$ has a transversal (Wanless' conjecture for odd-dimensional Latin hypercubes of order $4$). We prove that the number of perfect matchings of the bipartite hypergraph associated to an even-dimensional polystochastic $(0,1)$-tensor of order $4$ is positive. Furthermore, we extend some results concerning polystochastic $(0,1)$-tensors to nonnegative polystochastic tensors. Moreover, we prove that the permanent of a $ 4 $-dimensional nonnegative polystochastic tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. More generally, we show that the permanent of an even-dimensional nonnegative polystochastic tensor of order $n$ constructed using the row-Latin rectangle $R$ is positive. The result obtained here proves that the permanent of an even-dimensional nonnegative polystochastic tensor of order $4$ is positive.
We give a computer-assisted proof that if $G$ is a finite group of order $8pq$, where $p$ and $q$ are distinct primes, then every connected Cayley graph on $G$ has a hamiltonian cycle.
In a recent article, it was shown that the Heesch number in $\mathbb{E}^d$ is asymptotically unbounded for $d\to\infty$, by showing that, for each $d$ of the form $2^k$, there exists a hypersolid in $\mathbb{E}^d$ whose Heesch number equals $d-1$. We here show that the same holds not only for $d$ of the form $2^k$, but for any $d$, $d\geqslant 2$.
A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XY-plane that consists of up steps $\mathrm{u}=(1, 1)$, horizontal steps $\mathrm{h}=(1, 0)$, vertical steps $\mathrm{v}=(0, -1)$ and down steps $\mathrm{d}=(1, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathrm{u}$-steps, $\mathrm{h}$-steps, $\mathrm{v}$-steps and $\mathrm{d}$-steps are weighted respectively by $1, a, b$ and $c$.Let $\tau$ be a word on $\{\mathrm{u}, \mathrm{h}, \mathrm{v}, \mathrm{d}\}$, denote by $\mathcal{G}_n^{\tau}(a,b,c)$ the set of $\tau$-avoiding $(a,b,c)$-G-Motzkin paths of length $n$ for a pattern $\tau$. In this paper, we consider the $\mathrm{uvv}$-avoiding $(a,b,c)$-G-Motzkin paths and provide a direct bijection $\sigma$ between $\mathcal{G}_n^{\mathrm{uvv}}(a,b,b^2)$ and $\mathcal{G}_n^{\mathrm{uvu}}(a,b,b^2)$. Finally, the set of fixed points of $\sigma$ is also described and counted.
