arXiv subjects

Shikang Yu

Publications and source records attributed to Shikang Yu.

4 recordsLinked to original sources

arXiv2026

On the completion of $\epsilon$-dense partial Latin squares

A partial Latin square of order $n$ is called $\epsilon$-dense if each row and each column contains at most $\epsilon n$ filled cells, and each symbol occurs at most $\epsilon n$ times. A partial Latin square is said to be completable if its empty cells can be filled to obtain a Latin square. Daykin and H\"{a}ggkvist conjectured that every $\frac{1}{4}$-dense partial Latin square is completable. In this paper, we show that for all sufficiently large integers $n$, every $\frac{2}{25}$-dense partial Latin square of order $n$ is completable. The proof is obtained by establishing that there exists an $\eta > 0$ such that every triangle-divisible balanced tripartite graph on $3n$ vertices with partite minimum degree at least $(\frac{23}{25}-\eta)n$ admits a fractional triangle decomposition.

math.CO↗

arXiv2026

Fractional clique decompositions of dense balanced multipartite graphs

This paper concerns fractional $K_s$-decompositions of multipartite graphs. For integers $r\ge s\ge 3$, we consider balanced $r$-partite graphs $G$ on $rn$ vertices. We establish necessary conditions for $G$ to admit a fractional $K_s$-decomposition, extending the notion of $s$-admissibility from the case $r=s$ to $r>s$. Using an association scheme on the edge set of a complete $r$-partite graph, we prove that if $r\ge s+2$ and the partite minimum degree of $G$ is at least $(1-c)n$ with $c\le 1/((s-2)(s+1)(s-1)^4)$, then $G$ has a fractional $K_s$-decomposition. For $r=s+1$, we show that under the condition $c\le 1/(3s^3(s-2)^2)$, every $s$-admissible balanced $(s+1)$-partite graph with partite minimum degree at least $(1-c)n$ admits a fractional $K_s$-decomposition. These results provide new degree thresholds for fractional $K_s$-decompositions of multipartite graphs with more than $s$ parts.

math.CO↗

arXiv2026

Paley-type matrices and $1$-factorizations of complete graphs

Ball, Ortega--Moreno, and Prodromou asked two questions about whether, for every odd prime $p$, one can find a $1$-factor of the complete graph $K_{p+1}$ with some arithmetic restrictions related to quadratic residues. These problems are motivated by two natural compatibility conditions between $1$-factorizations and the sign patterns of certain Paley-type matrices. Recently, Afifurrahman et al. made some partial progress on the second problem. In this paper, we completely resolve both problems. We prove that the first problem has a solution precisely when $p\equiv3\pmod4$, while the second problem has a solution for every odd prime $p$. We also solve a further problem of Ball et al. for cyclic groups of odd order, and more generally for all finite abelian groups of odd order.

math.CO↗

arXiv2024

Existence of magic rectangle sets over finite abelian groups

Let $a$, $b$ and $c$ be positive integers. Let $(G,+)$ be a finite abelian group of order $abc$. A $G$-magic rectangle set MRS$_G(a,b;c)$ is a collection of $c$ arrays of size $a\times b$ whose entries are elements of a group $G$, each appearing exactly once, such that the sum of each row in every array equals a constant $\gamma\in G$ and the sum of each column in every array equals a constant $\delta\in G$. This paper establishes the necessary and sufficient conditions for the existence of an MRS$_G(a,b;c)$ for any finite abelian group $G$, thereby confirming a conjecture presented by Cichacz and Hinc.

math.CO↗