Dedicated ro the memory of Stan Ulam (31. ABSTRACTThe reconstruction number of graph G is the minimum number of pointdeleted subgraphs required in order to uniquely identify the original graph G. We list, based on computer calculations, the reconstruction number for all graphs with at most seven points. Some constructions and conjectures for graphs of higher order are given. The most striking statement is our concluding conjecture that almost all graphs have reconstruction number three.

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We say that a graphical invariant i of a graph interpolates over a family 8 of graphs if i satisfies the following property: If rn and M are the minimum and maximum values (respectively) of i over all graphs in 8 then for each k , rn 4 k I M , there is a graph H in 8 for which i ( H ) = k . In previous works it was shown that when 8 is the set of spanning trees of a connected graph G, a large number of invariants interpolate (some of these invariants require the additional assumption that G be 2-connected). Although the proofs of all these results use the same basic idea of gradually transforming one tree into another via a sequence of edge exchanges, some of these processes require sequences that use more properties of trees than do others. We show that the edge exchange proofs can be divided into three types, in accordance with the extent to which the exchange sequence depends upon properties of spanning trees. This idea is then used to obtain new interpolation results for some invariants, and to show how the exchange 'methods and interpolation results on spanning trees can be extended to other families of spanning subgraphs.

Chetwynd and Hilton showed that any regular graph G of even order n which has relatively high degree Á(G ) ! (( ffiffiffi 7 p À 1)=2) n has a 1-factorization. This is equivalent to saying that under these conditions G has chromatic index equal to its maximum degree Á(G ). Using this result, we show that any (not necessarily regular) graph G of even order n that has sufficiently high minimum degree (G ) ! ( ffiffiffi 7 p =3) n has chromatic index equal to its maximum degree providing that G does not contain an ''overfull'' subgraph, that is, a subgraph which trivially forces the chromatic index to be more than the maximum degree. This result thus verifies the Overfull Conjecture for graphs of even order and sufficiently high minimum degree. ß

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