arXiv subjects

Zhaochen Ding

Publications and source records attributed to Zhaochen Ding.

5 recordsLinked to original sources

arXiv2026

A classification of rotary embeddings of multicycles

We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle $\mathrm{C}_n^{(\lambda)}$ of length $n$ and edge-multiplicity $\lambda$, we determine all rotary embeddings for $n\geqslant 3$ and $\lambda\geqslant 2$. When $n$ is odd, there is a unique isomorphism class; when $n$ is even, the embeddings form a family $\mathcal{M}_n^{(\lambda)}(i,j)$ parameterized by integer pairs $(i,j)$ satisfying explicit congruence conditions.

math.CO↗

arXiv2025

Some new compatible groups

Two finite groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. We prove a new sufficient condition for two groups to be compatible. As a corollary, we obtain that nilpotent groups of the same order are compatible, and so are groups of the same square-free order.

math.GR↗

arXiv2025

On Bi-rotary Maps of Negative Prime Power Euler Characteristic

A map is bi-orientable if it admits an assignment of local orientations to its vertices such that for every edge, the local orientations at its two endpoints are opposite. Such an assignment is called a bi-orientation of the map. A bi-orientable map is bi-rotary if its automorphism group contains an arc-regular subgroup that preserves the bi-orientation. In this paper, we characterize the automorphism group structure of bi-rotary maps whose Euler characteristic is a negative prime power.

math.GR↗

arXiv2025

Embedding a Praeger-Xu graph into a surface

Rotary maps (orientably regular maps) are highly symmetric graph embeddings on orientable surfaces. This paper classifies all rotary maps whose underlying graphs are Praeger-Xu graphs, denoted $\operatorname{C}(p,r,s)$, for any odd prime $p$ that does not divide $r$. Our main result establishes a one-to-one correspondence between the isomorphism classes of these maps and the multiplicity-free representations of the dihedral group $\operatorname{D}_{2r}$ over the finite field $\mathbb{F}_p$. This work extends a recent classification for the case where $p=2$.

math.CO↗

arXiv2025

Some necessary conditions for compatibility of groups

Two groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. In this paper, we give new necessary conditions for two groups to be compatible.

math.GR↗