Graph Bundles Research Articles

Recognizing weighted directed cartesian graph bundles

In this paper we show that methods for recognizing Cartesian graph bundles can be generalized to weighted digraphs. The main result is an algorithm which lists the sets of degenerate arcs for all representations of digraph as a weighted directed Cartesian graph bundle over simple base digraphs not containing transitive tournament on three vertices. Two main notions are used.The first one is the new relation ▫$\vec{\delta}^\ast$▫ defined among the arcs of a digraph as a weighted directed analogue of the well-known relation ▫$\delta^\ast$▫. The second one is the concept of half-convex subgraphs. A subgraph ▫$H$▫ is half-convex in ▫$G$▫ if any vertex ▫$x \in G \setminus H$▫ has at most one predecessor and at most one successor

Natural Isomorphism Classes of Cycle Permutation Graphs Over a Graph

A permutation graph over a graph was introduced by Lee and Sohn in [8] as a generalization of both a graph bundle over a graph and a standard permutation graph, and they gave a characterization of a natural isomorphism and an automorphism of a given permutation graph over a graph. In this paper, we enumerate the natural isomorphism classes of cycle permutation graphs over a graph.

Algorithm for Recognizing Cartesian Graph Bundles

Algorithm for Recognizing Cartesian Graph Bundles

Bipartite graph bundles with connected fibres

Let G be a finite connected simple graph. The isomorphism classes of graph bundles and graph coverings over G have been enumerated by Kwak and Lee. Recently, Archdeacon and others characterised bipartite coverings of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of G, when G is nonbipartite. In this paper, we characterise bipartite graph bundles over G and derive some enumeration formulas of the isomorphism classes of them when the fibre is a connected bipartite graph. As an application, we compute the exact numbers of the isomorphism classes of bipartite graph bundles over G when the fibre is the path Pn or the cycle Cn.

Recognizing Cartesian graph bundles

Graph bundles generalize the notion of covering graphs and graph products. In this paper we extend some of the methods for recognizing Cartesian product graphs to graph bundles. Two main notions are used. The first one is the well-known equivalence relation δ ★ defined on the edge-set of a graph. The second one is the concept of k-convex subgraphs. A subgraph H is k-convex in G, if for any two vertices x and y of distance d, d ⩽ k, each shortest path from x to y in G is contained entirely in H. The main result is an algorithm that finds a representation as a nontrivial Cartesian graph bundle for all graphs that are Cartesian graph bundles over a triangle-free simple base. The problem of recognizing graph bundles over a base containing triangles remains open.

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K 2n) of the cartesian product of any graphG and a complete graphK 2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK 2n. Furthermore, ifn ? 2i(G) then the isoperimetric number of any graph bundle overG with fibreK n is equal to the isoperimetric numberi(G) ofG.

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

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.

Characteristic polynomials of complement of graph bundles and their applications

We study the structure of the complement of an F-bundle over a graph G when it can be expressed as an F-bundle over another graph G. In particular, we compute the characteristie polynomial of the complement of an F-bundle when its voltages lie in and abelian subgroup Г of Aut(F) which acts freely and transitively on the vertices of the fiber F Some applications to regular embeddings of graph bundles into Euclidean spaces are also discussed

The chromatic numbers of graph bundles over cycles

Graph bundles generalize the notion of covering graphs and products of graphs. The chromatic numbers of product bundles with respect to the Cartesian, strong and tensor product whose base and fiber are cycles are determined.

Coloring graph bundles

AbstractGraph bundles generalize the notion of covering graphs and products of graphs. Several results about the chromatic numbers of graph bundles based on the Cartesian product, the strong product and the tensor product are presented. © 1995 John Wiley & Sons, Inc.

Characteristic polynomials of some weighted graph bundles and its application to links

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted ‐bundles over a weighted graph G? can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are G As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

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.

Isomorphism Classes of Graph Bundles

AbstractRecently, M. Hofmeister [4] counted all nonisomorphic double coverings of a graph by using its Ζ2 cohomology groups, and J. Kwak and J. Lee [5] did the same work for some finite-fold coverings. In this paper, we give an algebraic characterization of isomorphic graph bundles, from which we get a formula to count all nonisomorphic graph-bundles. Some applications to wheels are also discussed.

The Maximum Genus of Graph Bundles

The upper embeddability of Cartesian and strong graph bundles with non-trivial base and fibre is proved. A similar result is obtained for both versions of lexicographic bundles. As a corollary the upper embeddability of Cartesian, strong and lexicographic products is obtained. The results cannot be generalized to the Cartesian and strong bundles with discrete fibres, i.e., to covering graphs. In this case sharp upper and lower bounds for Betti defficiency are obtained.

Edge-colorability of graph bundles

The topological notion of a fibre bundle is a generalization both of a Cartesian product and of a covering space. A graph bundle is a combinatorial analog of a fibre bundle. Accordingly, it is a generalization both of a Cartesian product of two graphs and of a covering graph. A “total graph” X is formed from a “base graph” B and “fibre” F. The edge-colorability of X is studied in terms of B and F. In particular, it is proved that if a graph bundle with base B and fibre F satisfies at least one of the conditions: (i) B is of chromatic class 1 and Δ(B) > 0, or (ii) F is of chromatic class 1 and Δ(F) > 0, or (iii) B and F both contain a 1-factor, then its total graph X is of chromatic class 1.