We show that the cover-index of an infinite graph can be expressed in terms of colouring properties of its finite subgraphs when the minimum degree of the graph is finite. We prove that every simple graph with infinite minimum degreeδ contains a tree which is regular of degreeδ and use this to prove that every graph with minimum degreeδ can be decomposed intoδ mutually edge-disjoint spanning subgraphs without ioslated vertices. In particular, the cover-index of a graph equals the minimum degree, when this is infinite.