arXiv subjects

Luyi Liu

Publications and source records attributed to Luyi Liu.

6 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

A classification of vertex-reversing maps with Euler characteristic coprime to the edge number

An arc-regular map is \emph{vertex-reversing} if the automorphism group has dihedral vertex stabilizers. This paper classifies solvable vertex-reversing maps whose Euler characteristic is coprime to the edge number. The classification establishes that such maps fall into four families: $\D_{2n}$-maps, $(\D_{2m}\times\D_{2n})$-maps, $(\ZZ_{mn\ell}{:}\D_4)$-maps, and $(\ZZ_{3^f n}.\S_4)$-maps, with the parameters specified in the main theorem. Moreover, for each family, we provide an explicit formula for the Euler characteristic.

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↗

arXiv2024

Arc-transitive maps with edge number coprime to the Euler characteristic -- I

This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one establishes a framework and carries out the classification work for arc-transitive maps with solvable automorphism groups, which illustrates how the edge number impacts on the Euler characteristic for maps. The classification is involved with the constructions of various new and interesting arc-regular maps.

math.CO↗

arXiv2024

Finite imprimitive rank $3$ affine groups -- I

This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not $p$-local, which shows that such groups are very rare, namely, the two non-isomorphic groups of the form $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup are the only examples.

math.GR↗

arXiv2024

Arc-transitive maps with coprime Euler characteristic and edge number -- II

This is the second of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one carries out the classification work for arc-transitive maps with non-solvable automorphism groups, which together with the first one completes a description of arc-transitive maps with the Euler characteristic and the edge number coprime. The classification is involved with a construction of some new and interesting reversing maps.

math.CO↗