Discrete Mathematics

The Eulerian polynomials enumerate permutations according to their number of descents. We initiate the study of descent polynomials over Cayley permutations, which we call Caylerian polynomials. Some classical results are generalized by linking ...

The Erdős-Gyárfás conjecture asserts that every graph with minimum degree at least three has a cycle whose length is a power of 2. Let G be a graph with minimum degree at least 3. We show that if G contains no induced path of order 10, then G ...

The edge-face chromatic number χ e f ( G ) of a plane graph G is the least number of colors such that any two adjacent or incident elements in E ( G ) ∪ F ( G ) receive different colors. In 2005, Luo and Zhang proved that each 2-connected simple ...

For a digraph Γ, if F is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of Γ, then F is called the splitting field of Γ. The extension degree of F over the field of rational numbers Q is said ...

Given a set F of graphs, we call a copy of a graph in F an F-graph. The F-isolation number of a graph G, denoted by ι ( G , F ), is the size of a smallest subset D of the vertex set V ( G ) such that the closed neighbourhood of D intersects the ...

In this work we introduce new combinatorial objects called d–fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set r d ( n ) to be their counting ...

• MacMahon's partition analysis to find generating functions.
• Eulerian polynomials connect with partition diamonds.
• q-series techniques to prove partition congruences including Ramanujan congruences.
• Generalizations of classical ...

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set S of vertices of a graph G is a locating-total dominating set if every vertex of G has a neighbor in S, and if any two vertices outside S have distinct ...

A pair ( A , B ) of hypergraphs is called orthogonal if | a ∩ b | = 1 for every pair of edges a ∈ A , b ∈ B. An orthogonal pair of hypergraphs is called a loom if each of its two members is the set of minimum covers of the other. Looms appear ...

An association scheme is P-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate P-polynomial association schemes of type ( α , β ) were introduced by Bernard et al., and multivariate P-...

Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting (AEAQEC, for short) code, and gave some good AEAQEC codes. In this paper, we provide two methods of constructing AEAQEC codes by means of ...

Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Difference Distribution Table (DDT), the Feistel Boomerang Connectivity Table (FBCT), the Feistel Boomerang Difference ...

For a graph G, a subset S ⊆ V ( G ) is called a cycle isolating set of G if G − N [ D ] contains no cycle. The cycle isolation number of G, denoted by ι c ( G ), is the minimum cardinality of a cycle isolating set of G. Recently, Borg proved that ...

In this paper we study queen's graphs, which encode the moves by a queen on an n × m chess board, through the lens of chip-firing games. We prove that their gonality is equal to nm minus the independence number of the graph, and give a one-to-one ...

The minimal excludant (mex) of a partition was introduced by Grabner and Knopfmacher under the name ‘least gap’ and was recently revived by Andrews and Newman. It has been widely studied in recent years together with the complementary partition ...

The toughness t ( G ) of a non-complete graph G is defined as t ( G ) = min ⁡ { | S | c ( G − S ) } in which the minimum is taken over all proper sets S ⊂ G such that G − S is disconnected, where c ( G − S ) denotes the number of components of G −...

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum ...

Linear complementary pair (abbreviated to LCP) of codes were defined by Ngo et al. in 2015, and were proved that these pairs of codes can help to improve the security of the information processed by sensitive devices, especially against so-called ...

For any simple-root constacyclic code C over a finite field F q, as far as we know, the group G generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group Aut ( C ) of C. ...

Based on the definition of Hamiltonian cycles by Katona and Kierstead, we present a recursive construction of tight Hamiltonian decompositions of complete 3-uniform hypergraphs K n ( 3 ), and complete multipartite 3-uniform hypergraph K t ( n ) ( ...

For η ∈ S 3, let S n av ( η ) denote the set of permutations in S n that avoid the pattern η, and let E n av ( η ) denote the expectation with respect to the uniform probability measure on S n av ( η ). For n ≥ k ≥ 2 and τ ∈ S k av ( η ), let N n ...

Recently, Liu and Liu gave the Singleton bound for pure asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. They constructed three new families of AQEAEC codes by means of Vandermonde matrices, generalized Reed-Solomon (GRS) ...

Given a graph F, let L ( F ) be a fixed finite family of graphs consisting of a C 4 and some bipartite graphs relying on an s-partite subgraph partitioning of edges of F. Define ( m , n , a , b )-graph by an m × n bipartite graph with n ≥ m such ...

In 2020, the first author and Pan proved that every edge-primitive graph of valency 6 is 2-arc-transitive, and except the complete bipartite graph K 6 , 6, the automorphism group is almost simple, and they determined such graphs having a solvable ...

Let G = 〈 y , x | y 2 n = 1 , x 2 = y n , x y x − 1 = y ℓ 〉 be the non-split metacyclic group with ℓ 2 ≡ 1 ( mod 2 n ) and ℓ ≢ ± 1 , n + 1 ( mod 2 n ). In this paper, we obtain the exact values of small Davenport constant d ( G ), Gao constant E (...

Given two graphs G and H, a size Ramsey game is played on the edge set of K N. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of G or a blue copy of H as soon as ...

In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, ...

The famous partition theorem of Euler states that partitions of n into distinct parts are equinumerous with partitions of n into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of n with all parts ...