Discrete Mathematics

For a positive integer d ⩾ 2, a family F ⊆ ( [ n ] k ) is said to be d-wise intersecting if | F 1 ∩ F 2 ∩ … ∩ F d | ⩾ 1 for all F 1 , F 2 , … , F d ∈ F. A d-wise intersecting family F ⊆ ( [ n ] k ) is called maximal if F ∪ { A } is not d-wise ...

Recall that a subset X of a group G is ‘product-free’ if X 2 ∩ X = ∅, i.e. if x y ∉ X for all x , y ∈ X. Let G be a group definable in a distal structure. We prove there are constants c > 0 and δ ∈ ( 0 , 1 ) such that every finite subset X ⊆ G ...

We initiate the study of a quantitative measure for the failure of a binary operation to be commutative and associative. We call this measure the associative-commutative spectrum as it extends the associative spectrum (also known as the ...

We introduce a non-backtracking Laplace operator for graphs and we investigate its spectral properties. With the use of both theoretical and computational techniques, we show that the spectrum of this operator captures several structural ...

In “Infinite families of near MDS codes holding t-designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428”, Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding 2-designs or 3-designs. Up ...

Quite recently, Mneimneh introduced the remarkable result whereby ∑ k = 0 n H k ( n k ) p k ( 1 − p ) n − k = ∑ i = 1 n 1 − ( 1 − p ) i i for 0 ≤ p ≤ 1 as the main result of a 2023 Discrete Mathematics article, where H k = 1 + 1 2 + ⋯ + 1 k ...

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. In this paper, we get a formula for the signed enumeration of alternating descents and in our proof we need a signed convolution ...

Let T be a tournament with n vertices v 1 , … , v n. The skew-adjacency matrix of T is the n × n zero-diagonal matrix S = [ s i j ] in which s i j = − s j i = 1 if v i dominates v j. It is well-known that the determinant of S is zero or the ...

An Euler tour in a hypergraph is a closed walk that traverses each edge of the hypergraph exactly once, while an Euler family is a family of closed walks that jointly traverse each edge exactly once and cannot be concatenated. In this paper, we ...

An automorphism group of a design is called block-transitive if it acts transitively on the blocks. This paper gives a reduction for Steiner 3-designs admitting a block-transitive automorphism group. We prove that a block-transitive point-...

Magic squares and anti-magic squares are the important research objects in combinatorics, and they are also well used in constructing graph labelings. Suppose that n and d are two positive integers with d < n. An n × n array A based on X = { 0 , ...

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. ...

In this paper, we consider optimal 1-planar multigraphs, that is, n-vertex multigraph having exactly 4 n − 8 edges and a drawing on the sphere such that each edge crosses at most one other edge, and that the drawing has no 2-gonal face. We ...

We introduce the following generalization of set intersection via characteristic vectors: for n , q , s , t ≥ 1 a family F ⊆ { 0 , 1 , … , q } n of vectors is said to be s-sum t-intersecting if for any distinct x , y ∈ F there exist at least t ...

Let G be a flag-transitive automorphism group of a symmetric ( v , k , λ ) design D with k > λ ( λ − 2 ). O'Reilly Regueiro proved that if G is point-imprimitive, then D has parameters ( v , k , λ ) = ( λ 2 ( λ + 2 ) , λ ( λ + 1 ) , λ ). In the ...

A graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertex-set with two orbits of equal size. A bi-Cayley graph over a dicyclic group is called a bi-dicirculant. We give a ...

Let m 1 , m 2 , n be three positive integers, m 1 < m 2 ≤ n. A ( m 1 , m 2 )-near-Skolem sequence of order n with defects m 1 and m 2 is a sequence S = ( s 1 , s 2 , ⋯ , s 2 n − 4 ) of 2 n − 4 positive integers which satisfies the following two ...

In this article we provide a version of the Central Sets Theorem, which was originally introduced by H. Furstenberg and considered to be a joint extension of Finite Sum Theorem and van der Waerden's theorem. The original version of this theorem ...

A labeling scheme is a space-efficient data structure for encoding graphs from a particular graph class. The idea is to assign each vertex of a graph a short label s.t. adjacency of two vertices can be algorithmically determined from their ...

We briefly discuss counterexamples to the conjecture of Lepović (2010) [6]] that a walk-regular graph, which is cospectral to its complement, has to be strongly regular.

An r-regular graph is said to be an r-graph if | ∂ ( X ) | ≥ r for each odd set X ⊆ V ( G ), where | ∂ ( X ) | denotes the set of edges with precisely one end in X. Note that every connected bridgeless cubic graph is a 3-graph. The Berge ...

Hitomezashi, a form of traditional Japanese embroidery, gives rise to intricate arrangements of axis-parallel unit-length stitches in the plane. Pete studied these patterns in the context of percolation theory, and the first two authors recently ...

The scramble number of a graph provides a lower bound for gonality and an upper bound for treewidth, making it a graph invariant of interest. In this paper we study graphs of scramble number at most two, and give a classification of all such ...

We determine the weight spectra of the Reed-Muller codes R M ( m − 3 , m ) for m ≥ 6 and R M ( m − 4 , m ) for m ≥ 8. The technique used is induction on m, using that the sum of two weights in R M ( r − 1 , m − 1 ) is a weight in R M ( r , m ), ...

In this paper we study Waring numbers g R ( k ) for ( R , m ) a finite commutative local ring with identity and k ∈ N with ( k , | R | ) = 1. We first relate the Waring number g R ( k ) with the diameter of the Cayley graphs G R ( k ) = C a y ( R ...

This paper is devoted to the classification of flag-transitive 2-designs. We prove that if D is a nontrivial imcomplete 2-( v , k , λ ) design with ( r − λ , k ) = 1 and G ≤ A u t ( D ) is flag-transitive with alternating socle, then D is the ...

How to efficiently represent a graph in computer memory is a fundamental data structuring question. In the present paper, we address this question from a combinatorial point of view. A representation of an n-vertex graph G is called implicit if ...

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent positions, we prove a q-analogue of Carlitz-...

The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with a and b adjacent if a b = 0. We show that the class of zero-divisor graphs is universal, in the sense that ...