Double covers of graphs

Abstract

In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers p : G -»• £?" for a given graph G_ . A classification theorem for double cover projections is obtained, and it is shown that the w-dimensional octahedron graph K " plays the role of universal object.

Counting double covers of graphs

“…This formula is well-known (see [ 5 ] ) ; usually it is proved by elementary combinatorial arguments.…”

“…The relationship between coverings and automorphisms of graphs has been studied by several authors (for examples, see [ 2 ] and [3]); Waller [ 5 ] classified double covers by a category-theoretic approach. The purpose of the present paper is to count double covers with respect to r.…”

Counting some finite-fold coverings of a graph

“…Waller [4] and Hofmeister [3] counted the number of isomorphism classes of double coverings of a given graph G. It is natural to ask how many nonisomorphic n-fold coverings of G there are. In this paper, we give some partial solutions of this question.…”

“…In this paper, we give some partial solutions of this question. Coverings of G and their maps form a category, and if two coverings are isomorphic their fibres have the same cardinality; they are n-fold for the same n. If the condition p2 o ( = pl is replaced by p2 o ( = 7 o pl for an automorphism 7 of G, we say ~ covers 7; Hofmeister [3] allows isomorphisms covering general 7, and Waller [4], though requiring the diagram above, remarks that K4 has exactly three isomorphism classes of double coverings, which is true only with Hofmeister's definition.…”

Bipartite graph bundles with connected fibres

“…After the enumeration of double covers of a graph in [4] and [8], there has been much progress during the last decade in the enumeration of several graph coverings or graph bundles over a graph ([l, 4, 5, 6] and references there).…”