On some questions of Razpet regarding binomial coefficients

We answer two questions of Razpet (Discrete Math. 135 (1994) 377) regarding finite submatrices of the Pascal triangle. One of these has been solved independently in another way by Bayat and Teimoori (Linear Algebra Appl. 308 (2000) 65).     

On weighted q-Daehee polynomials with their applications

In this paper, we first consider a generalization of Kim’s $p$-adic $q$-integral on ${\mathbb{Z}}_p$ including parameters $\alpha$ and $\beta$. By using this integral, we introduce the $q$-Daehee polynomials and numbers with weight $\left(\alpha ,\beta \right)$. Then, we obtain some interesting relationships and identities for these numbers and polynomials. We also derive some correlations among $q$-Daehee polynomials with weight $\left(\alpha ,\beta \right)$, $q$-Bernoulli polynomials with weight $\left(\alpha ,\beta \right)$ and Stirling numbers of second kind.     

On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.     

On the smallest non-trivial tight sets in Hermitian polar spaces H(d,q2), d even

We show that every $x$-tight set of a Hermitian polar spaces $\mathrm{H}(2n,q^2)$, $n\ge 2$, is the union of $x$ disjoint generators of the polar space provided that $x\le \frac{1}{2}(q+1)$. This was known before only when $n\in \{2,3\}$. This result is a contribution to the conjecture that the smallest $x$-tight set of $H(2n,q^2)$ that is not a union of disjoint generators occurs for $x=q+1$ and is for sufficiently large $q$ an embedded subgeometry.     

On two types of Z-monodromy in triangulations of surfaces

Let $\Gamma$ be a triangulation of a connected closed 2-dimensional (not necessarily orientable) surface. Using zigzags (closed left–right walks), for every face of $\Gamma$ we define the $z$-monodromy which acts on the oriented edges of this face. There are precisely 7 types of $z$-monodromies. We consider the following two cases: (M1) the $z$-monodromy is identity, (M2) the $z$-monodromy is the consecutive passing of the oriented edges. Our main result is the following: the subgraphs of the dual graph ${\Gamma }^1$ formed by edges whose $z$-monodromies are of types (M1) and (M2), respectively, both are forests. We apply this statement to the connected sum of $z$-knotted triangulations.     

On cycles in regular 3-partite tournaments

A $c$-partite tournament is an orientation of a complete $c$-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament $D$ is contained in an $m$-, $(m+1)$- or $(m+2)$-cycle for each $m\in \{3,4,\dots ,\left|V\left(D\right)\right|?2\}$, and this conjecture was proved to be correct for $3\le m\le 7$. In 2012, Xu et al. conjectured that every arc of an $r$-regular 3-partite tournament $D$ with $r\ge 2$ is contained in a $(3k?1)$- or $3k$-cycle for $k=2,3,\dots ,r$. They proved that this conjecture is true for $k=2$. In this paper, we confirm this conjecture for $k=3$, which also implies that Volkmann’s conjecture is correct for $m=7,8$.     

On the classification of linear complementary dual codes

We give a complete classification of binary linear complementary dual codes of lengths up to 13 and ternary linear complementary dual codes of lengths up to 10.     

On circular–linear one-factorizations of the complete graph

In Korchmáros et al. (2018)one-factorizations of the complete graph $K_n$ are constructed for $n=q+1$ with any odd prime power $q$ such that either $q\equiv 1(mod4)$ or $q=2^h?1$. The arithmetic restriction $n=q+1$ is due to the fact that the vertices of $K_n$ in the construction are the points of a conic $\Omega$ in the finite plane of order $q$. Here we work on the Euclidean plane and describe an analogous construction where the role of $\Omega$ is taken by a regular $n$-gon. This allows us to remove the above constraints and construct one-factorizations of $K_n$ for every even $n\ge 6$.     

Identities from representation theory

We give a new Jacobi–Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, a slight modification of the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard’s identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.     

On edge-primitive graphs of order twice a prime power

A graph is called edge-primitive if its automorphism group acts primitively on its edge set. In this paper, edge-primitive graphs of order twice a prime power are completely determined.     

Quasisymmetric and Schur expansions of cycle index polynomials

Given a subgroup $G$ of the symmetric group $S_n$, the cycle index polynomial ${\mathrm{cyc}}_G$ is the average of the power-sum symmetric polynomials indexed by the cycle types of permutations in $G$. By Pólya’s Theorem, the monomial expansion of ${\mathrm{cyc}}_G$ is the generating function for weighted colorings of $n$ objects, where we identify colorings related by one of the symmetries in $G$. This paper develops combinatorial formulas for the fundamental quasisymmetric expansions and Schur expansions of certain cycle index polynomials. We give explicit bijective proofs based on standardization algorithms applied to equivalence classes of colorings. Subgroups studied here include Young subgroups of $S_n$, the alternating groups $A_n$, direct products, conjugate subgroups, and certain cyclic subgroups of $S_n$ generated by $(1,2,\dots ,k)$. The analysis of these cyclic subgroups when $k$ is prime reveals an unexpected connection to perfect matchings on a hypercube with certain vertices identified.     

On the asymptotic growth of bipartite graceful permutations

We give exact growth rates for the number of bipartite graceful permutations of the symbols $\{0,1,\dots ,n?1\}$ that start with $a$ for $a\le 14$ (equivalently, $\alpha$-labelings of paths with $n$ vertices that have $a$ as a pendant label). In particular, when $a=14$ the growth is asymptotically like ${\lambda }^n$ for $\lambda \approx 3.461$. The number of graceful permutations of length $n$ grows at least this fast, improving on the best existing asymptotic lower bound of $2.37^n$. Combined with existing theory, this improves the known lower bounds on the number of Hamiltonian decompositions of the complete graph $K_{2n+1}$ and on the number of cyclic oriented triangular embeddings of $K_{12s+3}$ and $K_{12s+7}$. We also give the first exponential lower bound on the number of R-sequencings of ${\mathbb{Z}}_{2n+1}$.     

On lifted cover inequalities: A new lifting procedure with unusual properties

Lifted cover inequalities are well-known cutting planes for 0–1 linear programs. We show how one of the earliest lifting procedures, due to Balas, can be significantly improved. The resulting procedure has some unusual properties. For example, (i) it can yield facet-defining inequalities even if the given cover is not minimal, (ii) it can yield facet-defining inequalities that cannot be obtained by standard lifting procedures, and (iii) the associated superadditive lifting function is integer-valued almost everywhere.     

On a characterization of path connected topological fields

The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space X and the space of maximal ideals of the ring of real valued continuous functions $C(X,\mathbb{R})$. More explicitly, our motivation is the following question: What is the essential property of the topological field $F=\mathbb{R}$ that makes such a correspondence valid for all compact Hausdorff spaces? It turns out that such a perfect correspondence exists if and only if F is a path connected topological field.     

On incidence choosability of cubic graphs

An incidence of a graph $G$ is a pair $(u,e)$ where $u$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $u$. Two incidences $(u,e)$ and $(v,f)$ of $G$ are adjacent whenever (i) $u=v$, or (ii) $e=f$, or (iii) $uv=e$ or $uv=f$. An incidence$k$-coloring of $G$ is a mapping from the set of incidences of $G$ to a set of $k$ colors such that every two adjacent incidences receive distinct colors. The notion of incidence coloring has been introduced by Brualdi and Quinn Massey (1993) from a relation to strong edge coloring, and since then, has attracted a lot of attention by many authors.On a list version of incidence coloring, it was shown by Benmedjdoub et al. (2017) that every Hamiltonian cubic graph is incidence 6-choosable. In this paper, we show that every cubic (loopless) multigraph is incidence 6-choosable. As a direct consequence, it implies that the list strong chromatic index of a $(2,3)$-bipartite graph is at most 6, where a (2,3)-bipartite graph is a bipartite graph such that one partite set has maximum degree at most 2 and the other partite set has maximum degree at most 3.     

On the reliability roots of simplicial complexes and matroids

Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ fail independently with probability $q\in [0,1]$. The all-terminal reliability of $G$ is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in $q$, and it was conjectured that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few and far between, and the roots are only barely outside the disk. In this paper we generalize the notion of reliability to simplicial complexes and matroids and investigate when the roots fall inside the closed unit disk. We show that such is the case for matroids of rank 3 and paving matroids of rank 4. We also prove that the reliability roots of shellable complexes are dense in the complex plane, and that the real reliability roots of any matroid lie in $[?1,0)\cup \left\{1\right\}$. Finally, we also show that the reliability roots of thickenings of the Fano matroid can lie outside the unit disk.     

On Nordhaus–Gaddum type inequalities for the game chromatic and game coloring numbers

A seminal result by Nordhaus and Gaddum states that $2\sqrt{n}\le \chi \left(G\right)+\chi \left(\overline{G}\right)\le n+1$ for every graph $G$ of order $n$, where $\overline{G}$ is the complement of $G$ and $\chi$ is the chromatic number. We study similar inequalities for ${\chi }_g\left(G\right)$ and $co{l}_g\left(G\right)$, which denote, respectively, the game chromatic number and the game coloring number of $G$. Those graph invariants give the score for, respectively, the coloring and marking games on $G$ when both players use their best strategies.     

On the (3,1)-choosability of planar graphs without adjacent cycles of length 5,6,7

A $(k,d)$-list assignment $L$ of a graph $G$ is a mapping that assigns to each vertex $v$ a list $L\left(v\right)$ of at least $k$ colors satisfying $|L\left(x\right)\cap L\left(y\right)|\le d$ for each edge $xy$. A graph $G$ is $(k,d)$-choosable if there exists an $L$-coloring of $G$ for every $(k,d)$-list assignment $L$. This concept is also known as choosability with separation. In this paper, we prove that any planar graph $G$ is $(3,1)$-choosable if any $i$-cycle is not adjacent to a $j$-cycle, where $5\le i\le 6$ and $5\le j\le 7$.