Jin Ho Kwak

2-Arc-transitive regular covers of complete graphs Having the covering transformation group Z p 3

A family of 2-arc-transitive regular covers of a complete graph is investigated. In this paper, we classify all such covering graphs satisfying the following two properties: (1) the covering transformation group is isomorphic to the elementary abelian p-group Zp3, and (2) the group of fiber-preserving automorphisms acts 2-arc-transitively. As a result, new infinite families of 2-arc-transitive graphs are constructed.

One-regular cubic graphs of order a small number times a prime or a prime square

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2 p or 2 p 2 where p is a prime if and only if 3 is a divisor of p 1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4 p or 4 p 2 .

Total embedding distributions for bouquets of circles

Crosscap-number distributions, the distribution of graph embeddings into nonorientable surfaces, have been known for only a few cases. Chen et al. (Discrete Math. 128 (1994) 73) calculated the crosscap-number distribution of necklaces, closed-end ladders and cobblestone paths. In this paper, we compute the total genus polynomials and the total embedding polynomials of bouquets of circles with an aid of edge-attaching surgery technique. It extends their genus distributions computed by Gross et al. (J. Combin. Theory (B) 47 (1989) 292). The same work is also done for dipoles.

Regular graph coverings whose covering transformation groups have the isomorphism extension property

Abstract Enumerative results are presently a major center of interest in topological graph theory, as in the work of Gross and Furst [1], Hofmeister [5,6], Kwak and Lee [9–13] and Mull et al. [15], etc. Kwak and Lee [9] enumerated the isomorphism classes of graph bundles and those of n -fold graph coverings with respect to a group of automorphisms of the base graph which fix a spanning tree. Hofmeister [6] enumerated independently the isomorphism classes of n -fold graph coverings with respect to the trivial automorphism group of the base graph. But the enumeration of isomorphism classes of regular graph coverings has not been answered completely. As its partial answers, Hofmeister enumerated the isomorphism classes of Z 2 -coverings (double coverings) with respect to any group of automorphisms of the base graph, and Sato [14] did the same work for Z p -coverings (regular prime-fold coverings). With respect to the trivial automorphism group of the base graph, Hong and Kwak [8] did the same work for Z 2 ⊕ Z 2 or Z 4 -coverings, and Kwak and Lee [10] did it for Z p , Z p ⊕ Z 1 ( p ≠ q primes) or Z p 2 -coverings. As an expansion of this effort, we obtain in this paper several new algebraic characterizations for isomorphic regular coverings and derive an enumerating formula for the isomorphism classes of A-coverings of a graph G with respect to any group of automorphisms of G which fix a spanning tree, when the covering transformation group A has the isomorphism extension property. By definition, it means that every isomorphism between any two isomorphic subgroups B 1 and B 2 of A can be extended to an automorphism of A. Also, we obtain complete numerical enumeration of the isomorphism classes of Z n -coverings for all n , D n -coverings for odd n (D n is the dihedral group of order 2 n ) or Z p ⊕ Z p -coverings of a graph G for prime p with respect to the trivial automorphism group of G . In addition, we applied our results to a bouquet of circles.

Tetravalent half-arc-transitive graphs of order p4

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. It is known that for a prime p there is no tetravalent half-arc-transitive graphs of order p or p^2. Xu [M.Y. Xu, Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified the tetravalent half-arc-transitive graphs of order p^3. As a continuation, we classify in this paper the tetravalent half-arc-transitive graphs of order p^4. It shows that there are exactly p-1 nonisomorphic connected tetravalent half-arc-transitive graphs of order p^4 for each odd prime p.

Enumeration of graph embeddings

Abstract For a finite connected simple graph G , let Γ be a group of graph automorphisms of G . Two 2-cell embeddings ι: G → S and j : G → S of a graph G into a closed surface S (orientable or nonorientable) are congruent with respect to Γ if there are a surface homeomorphism h : S → S and a graph automorphism γϵΓ such that h oι= j oγ. In this paper, we give an algebraic characterization of congruent 2-cell embeddings, from which we enumerate the congruence classes of 2-cell embeddings of a graph G into closed surfaces with respect to a group of automorphisms of G , not just the full automorphism group. Some applications to complete graphs are also discussed. As an orientable case, the oriented congruence of a graph G into orientable surfaces with respect to the full automorphism group of G was enumerated by Mull et al. (1988).

A classification of regular t-balanced Cayley maps on dicyclic groups

In their study of regular t-balanced Cayley maps on a group, Conder et al. [M. Conder, R. Jajcay, T. Tucker, Regular t-balanced Cayley maps, J. Combin. Theory Ser. B (in press)] recently classified the regular anti-balanced Cayley maps on an abelian group. In this paper, we classify the regular t-balanced Cayley maps on a dihedral group for any t .

Regular embeddings of complete multipartite graphs

In this paper, we classify all regular embeddings of the complete multipartite graphs Kp,...,p for a prime p into orientable surfaces. Also, the same work is done for the regular embeddings of the lexicographical product of any connected arc-transitive graph of prime order q with the complement of the complete graph of prime order p, where q and p are not necessarily distinct. Lots of regular maps found in this paper are Cayley maps.

Linear criteria for lifting automorphisms of elementary abelian regular coverings

Abstract For a given finite connected graph Γ , a group H of automorphisms of Γ and a finite group A , a natural question can be raised as follows: Find all the connected regular coverings of Γ having A as its covering transformation group, on which each automorphism in H can be lifted. In this paper, we investigate the regular coverings with A= Z p n , an elementary abelian group and get some new matrix-theoretical characterizations for an automorphism of the base graph to be lifted. As one of its applications, we classify all the connected regular covering graphs of the Petersen graph satisfying the following two properties: (1) the covering transformation group is isomorphic to the elementary abelian p -group Z p n , and (2) the group of fibre-preserving automorphisms of a covering graph acts arc-transitively. As a byproduct, some new 2- and 3-arc-transitive graphs are constructed.

Caracteristics polynomials of some grap bundlesII

The characteristic polynomial of a graph G is that of its adjacency matrix, and its eigenvalues are those of its adjacency matrix. Recently, Y. Chae, J. H. Kwak and J. Lee showed a relation between the characteristic polynomial of a graph G and those of graph bundles over G. In particular, the characteristic polynomial of G is a divisor of those of its covering graphs. They also gave the complete computation of the characteristic polynomials of K 2 ( )-bundles over a graph. In this paper, we compute the characteristic polynomial of a graph bundle when its voltages lie in an abelian subgroup of the full automorphism group of the fibre; in particular, the automorphism group of the fibre is abelian. Some applications to path- or cycle-bundles are also discussed.