Discrete Mathematics最新文献

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 $\mathrm{Aut}\left(C\right)$ of $C$. In this paper, by calculating the number of $G$-orbits of $C﹨\left\{0\right\}$, we give an explicit upper bound on the number of non-zero weights of $C$ and present a necessary and sufficient condition for $C$ to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in Zhang and Cao (2024) [26]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code $C$ belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of $C$ by substituting $G$ with a larger subgroup of $\mathrm{Aut}\left(C\right)$. The results derived in this paper generalize the main results in Chen et al. (2024) [9].

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 statistic maximal excludant (maex), first introduced by Chern. Among such recent works, the first and second authors along with Maji introduced and studied the r-chain minimal excludants (r-chain mex) which led to a new generalization of Euler's classical partition theorem and the sum-of-mex identity of Andrews and Newman. In this paper, we first give combinatorial proofs for these two results on r-chain mex. Then we also establish the associated identity for the r-chain maximal excludant, recently introduced by the first two authors and Maji, both analytically and combinatorially.

For a graph G, a subset $S\subseteq V\left(G\right)$ is called a cycle isolating set of G if $G-N\left[D\right]$ contains no cycle. The cycle isolation number of G, denoted by ${\iota }_c\left(G\right)$, is the minimum cardinality of a cycle isolating set of G. Recently, Borg proved that if G is a connected n-vertex graph that is not a triangle, then ${\iota }_c\left(G\right)\le \frac{n}{4}$. In this paper, we prove that if G is a connected triangle-free n-vertex graph that is not a 4-cycle, then ${\iota }_c\left(G\right)\le \frac{n}{5}$. In particular, we characterize the subcubic graphs that attain the bound. For graphs with larger girth, several conjectures are proposed.

The toughness $t\left(G\right)$ of a non-complete graph G is defined as $t\left(G\right)=\min \left\{\frac{\left|S\right|}{c(G-S)}\right\}$ in which the minimum is taken over all proper sets $S\subset G$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. Conjectured by Brouwer and proved by Gu, a toughness theorem state that every d-regular connected graph always has $t\left(G\right)>\frac{d}{\lambda }-1$, where λ is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a variation of toughness $\tau \left(G\right)$ of a graph G, which is defined by $\tau \left(G\right)=\min \{\frac{\left|S\right|}{c(G-S)-1},S\subset V(G\left)\text{and}c\right(G-S)>1\}$. By incorporating the variation of toughness and spectral conditions, we provide spectral conditions for a graph to be τ-tough ($\tau \ge 2$ is an integer) and to be τ-tough ($\frac{1}{\tau }$ is a positive integer) with minimum degree δ, respectively. Additionally, we also investigate a analogous problem concerning balanced bipartite graphs.

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 Table (FBDT) and the Feistel Boomerang Extended Table (FBET) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, the results on them are rare. In this paper, we investigate the properties of the power function $F\left(x\right):=x^{2^{m+1}-1}$ over the finite field $F_{2^n}$ of order $2^n$ where $n=2m$ or $n=2m+1$ (m stands for a positive integer). As a consequence, by carrying out certain finer manipulations of solving specific equations over $F_{2^n}$, we give explicit values of all entries of the DDT, the FBCT, the FBDT and the FBET of the investigated power functions. From the theoretical point of view, our study pushes further former investigations on differential and Feistel boomerang differential uniformities for a novel power function F. From a cryptographic point of view, when considering Feistel block cipher involving F, our in-depth analysis helps select F resistant to differential attacks, Feistel differential attacks and Feistel boomerang attacks, respectively.

In this paper we study queen's graphs, which encode the moves by a queen on an $n\times m$ 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 correspondence between maximum independent sets and classes of positive rank divisors achieving gonality. We also prove an identical result for toroidal queen's graphs.

A connected graph $\Gamma =(V,E)$ is called a basic 2-arc-transitive graph if its full automorphism group has a 2-arc-transitive subgroup G, and every minimal normal subgroup of G has at most two orbits on V. In 1993, Praeger proved that every finite 2-arc-transitive connected graph is a cover of some basic 2-arc-transitive graph, and proposed the classification problem of finite basic 2-arc-transitive graphs. In this paper, a classification is given for basic 2-arc-transitive non-bipartite graphs of order $r^a{s}^b$ and basic 2-arc-transitive bipartite graphs of order $2r^a{s}^b$, where r and s are distinct primes.

In 2013, Ku and Wong showed that for any partitions μ and ${\mu }^{\prime }$ of a positive integer n with the same first part u and the lexicographic order $\mu ◃{\mu }^{\prime }$, the eigenvalues ${\xi }_{\mu }$ and ${\xi }_{{\mu }^{\prime }}$ of the derangement graph ${\Gamma }_n$ have the property $\left|{\xi }_{\mu }\right|\le \left|{\xi }_{{\mu }^{\prime }}\right|$, where the equality holds if and only if $u=3$ and all other parts are less than 3. In this article, we obtain an analogous conclusion on the eigenvalues of the perfect matching derangement graph $M_{2n}$ of $K_{2n}$ by finding a new recurrence formula for the eigenvalues of $M_{2n}$.

In this paper we consider the seating couples problem with an even number of seats, which, using graph theory terminology, can be stated as follows. Given a positive even integer $v=2n$ and a list L containing n positive integers not exceeding n, is it always possible to find a perfect matching of $K_v$ whose list of edge-lengths is L? Up to now a (non-constructive) solution is known only when all the edge-lengths are coprime with v. In this paper we firstly present some necessary conditions for the existence of a solution. Then, we give a complete constructive solution when the list consists of one or two distinct elements, and when the list consists of consecutive integers $1,2,\dots ,x$, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.

A permutation of the integers avoiding monotone arithmetic progressions of length 6 was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length 5. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length 4. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length 5. A permutation of the positive integers that avoided monotone arithmetic progressions of length 4 with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\ge 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length 4 with common difference not divisible by $2^k$. In addition, we specify the structure of permutations of $[1,n]$ that avoid length 3 monotone arithmetic progressions mod n as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length k monotone arithmetic progressions mod n.