A month or two ago, someone (Ed Pegg Jr. ?) posed a problem I'm calling "domino network". Surely this isn't new, but I didn't find any mention via some web searches. Anyone know of a reference? Problem: Label the nodes of a 4x7 rectangular graph with 0 through 9 (repeats allowed), such that each of the 45 edges corresponds to a different domino in the set of 45 dominoes with 0 (blank) through 9 pips, without doubles. The above problem is impossible (sketch below). I'm looking into related problems: varying the pip limit, whether doubles are included or not, and using various simple shapes on a squre, triangular, or hexagonal grid. Thanks for any comments! -- Mike Beeler Proof by contradiction. Suppose there is a solution. Consider the set of nodes labeled "0". Together, they must connect to 9 other nodes: once to a "1", once to a "2", etc. This is true of the nodes with any given label. So, the solution has 10 sets of nodes, each set connecting to exactly 9 other nodes. The given graph has 4 nodes (the corners) with valence 2; 14 nodes (sides) of valence 3; and 10 of valence 4. There is no way to partition these 28 nodes into 10 sets with the valence sum in each set being 9. I'm calling these "domino networks" because "domino graph" seems to already mean other things. (A particular 6-node graph, per MathWorld; or, a graph with a domino at each node, and links between dominos that have no number of pips in common, per the maa.org page by Ed. There is IBM Lotus Domino, a server product, so "domino network" is unfortunately a bit ambiguous too.)