Discrete Mathematics最新文献

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 existence of fractional revival on circulant graphs and cubelike graphs are characterized.

In this paper, we study 2-designs $D=(P,B^{S_n})$, where $P$ can be viewed as the edge set of the complete graph $K_n$, and B can be identified as the edge set of a subgraph of $K_n$. We give a necessary condition for $S_n$ to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group $S_n$. As an application, all pairs $(D,G)$ are determined, where $D$ is a 2-$(v,k,\lambda )$ design with $\gcd (v-1,k-1)=3$ or 4, and G is flag-transitive with $Soc\left(G\right)=A_n$ for $n\ge 5$. Furthermore, we show that there are infinite flag-transitive, point-primitive 2-$(v,k,\lambda )$ designs with $\gcd (v-1,k-1)\le {(v-1)}^{1/2}$ and alternating socle $A_n$ with $v=\left(\begin{array}{c}n\\ 2\end{array}\right)$.

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 repeated at least once equals the number of partitions of n where all parts must be even or congruent to $3\left(\mathrm{mod}6\right)$. These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese Remainder Theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function $p\left(n\right)$ and related partition functions.

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 possible. The online (size) Ramsey number $\tilde{r}(G,H)$ is the number of rounds in the game provided Builder and Painter play optimally. We prove that $\tilde{r}(C_4,P_n)\le 2n-2$ for every $n\ge 8$. The upper bound matches the lower bound obtained by J. Cyman, T. Dzido, J. Lapinskas, and A. Lo, so we get $\tilde{r}(C_4,P_n)=2n-2$ for $n\ge 8$. Our proof for $n\le 13$ is computer-assisted. The bound $\tilde{r}(C_4,P_n)\le 2n-2$ solves also the “all cycles vs. $P_n$” game for $n\ge 8$ – it implies that it takes Builder $2n-2$ rounds to force Painter to create a blue path on n vertices or any red cycle.

In this paper, we continue to study the existence of large sets plus of partitioned incomplete Latin squares of type $g^n{\left(ug\right)}^1$, denoted by LSPILS${}^{+}\left(g^n{\left(ug\right)}^1\right)$. We almost solve the existence of an LSPILS${}^{+}\left(g^n{\left(ug\right)}^1\right)$ for any integer $g\ge 1$ and $u=3,4$ with some possible exceptions.

Let $F$ be a family of r-uniform hypergraphs, and let H be an r-uniform hypergraph. Then H is called $F$-free if it does not contain any member of $F$ as a subhypergraph. The Turán number of $F$, denoted by ${\mathrm{ex}}_r(n,F)$, is the maximum number of hyperedges in an $F$-free n-vertex r-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický et al. (2013) [17] determined the Turán number $\mathrm{ex}(n,F)$ of a star forest F for sufficiently large n. Recently, Khormali and Palmer (2022) [13] generalized the above result to three different well-studied hypergraph settings (the expansions of a graph, linear hypergraphs and Berge hypergraphs), but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general star forests in hypergraphs.

We study the Vapnik–Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $F_q$, when considered as a subset of the additive group. We conjecture that as $q\to \infty$, the squares have the maximum possible VC-dimension, viz. $(1+o(1\left)\right){\log }_{2}q$. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is $⩾(\frac{1}{2}+o(1\left)\right){\log }_{2}q$. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups $\Gamma \subseteq F_q^{\times }$ of bounded index.

Let $G=〈y,x|y^{2n}=1,x^2=y^n,xy{x}^{-1}=y^{\ell }〉$ be the non-split metacyclic group with ${\ell }^2\equiv 1\left(\mathrm{mod}2n\right)$ and $\ell \not\equiv \pm 1,n+1\left(\mathrm{mod}2n\right)$. In this paper, we obtain the exact values of small Davenport constant $d\left(G\right)$, Gao constant $E\left(G\right)$, η-constant $\eta \left(G\right)$ and Erdős-Ginzburg-Ziv constant $s\left(G\right)$. Additionally, we study the associated inverse problems on $d\left(G\right)$, $E\left(G\right)$, $\eta \left(G\right)$ and $s\left(G\right)$. In 2003, Gao conjectured that $s\left(G\right)=\eta \left(G\right)+\text{exp}\left(G\right)-1$ for any finite group G. In 2005, Gao and Zhuang conjectured that $E\left(G\right)=d\left(G\right)+\left|G\right|$ for any finite group G. As a result, we confirm the two conjectures for non-split metacyclic groups.

A cut-down de Bruijn sequence is a cyclic string of length L, where $1\le L\le k^n$, such that every substring of length n appears at most once. Etzion [Theor. Comp. Sci 44 (1986)] introduced an algorithm to construct binary cut-down de Bruijn sequences requiring $o\left(n\right)$ simple n-bit operations per symbol generated. In this paper, we simplify the algorithm and improve the running time to $O\left(n\right)$ time per symbol generated using $O\left(n\right)$ space. Additionally, we develop the first successor-rule approach for constructing a binary cut-down de Bruijn sequence by leveraging recent ranking/unranking algorithms for fixed-density Lyndon words. Finally, we develop an algorithm to generate cut-down de Bruijn sequences for $k>2$ that runs in $O\left(n\right)$ time per symbol using $O\left(n\right)$ space after some initialization.