Decompositions of infinite graphs: I—bond-faithful decompositions

Abstract

We show that a graph can always be decomposed into edge-disjoint subgraphs of countable cardinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinal. We also show that this result, with the assumption of the Generalized Continuum Hypothesis, can be generalized to any uncountable cardinal. As applications of such decompositions we prove some results about Seymour's double-cover conjecture for infinite graphs, and about the maximal number of edge-disjoint spanning trees in graphs having high edge-connectivity. However, the main motivation for introducing these decompositions can be found in the second part of this paper where, to achieve a complete solution of the circuit decomposition problem (i.e. the problem of characterizing the graphs that admit decompositions into 2-regular connected subgraphs), we use the results of this first part to carry out a reduction to the countable case.