arXiv subjects

Jin Ho Kwak

Publications and source records attributed to Jin Ho Kwak.

4 recordsLinked to original sources

arXiv2017

On basic graphs of symmetric graphs of valency five

A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient graph $\G_N$ has the same valency with $\G$. In this paper, we classify symmetric basic graphs of order $2qp^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with $5\di (q-1)$, the complete graph $K_6$ of order $6$, the complete bipartite graph $K_{5,5}$ of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order $kp^n$ for some small integers $k$ and $n$ are classified.

math.CO↗

arXiv2015

Lifting graph automorphisms along solvable regular covers

A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of lifts of the automorphism in the layers of the abelian covers. This procedure is applied to classify metacyclic covers of the tetrahedron branched at face-centers.

math.GT↗

arXiv2011

2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs

We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B, transposed by an automorphism of order 2. The case where G is metacyclic having been dealt with elsewhere, we show that for each e>2 there are exactly three such non-metacyclic groups G with $|A|=|B|=2^e$, and for e=2 there is one. These groups appear in a classification by Berkovich and Janko of 2-groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.

math.GR↗

arXiv2010

Classification of nonorientable regular embeddings of complete bipartite graphs

A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs $K_{n,n}$ into nonorientable surfaces. Such regular embedding of $K_{n,n}$ exists only when $n = 2p_1^{a_1}p_2^{a_2}... p_k^{a_k}$ (a prime decomposition of $n$) and all $p_i \equiv \pm 1 (\mod 8)$. In this case, the number of those regular embeddings of $K_{n,n}$ up to isomorphism is $2^k$.

math.CO↗