Arch. Math., Vol. 65, 461-464 (1995) 0003-889X/95/6505-0461 $ 2.30/0 9 1995 Birkhfiuser Verlag, Basel An application of coding theory to a problem in graphical enumeration By DIETER JUNGNICKELand SCOTT A. VANSTONE *) In this note we exploit a relationship between graph theory and coding theory to obtain a very short and elegant proof of Read's theorem giving the generating function for the number of Eulerian graphs with p vertices and an analogous (to our knowledge new) theorem concerning bipartite Eulerian graphs. Let us recall the necessary back- ground. Let G = (V, E) be a finite graph with vertex set V and edge set E. We let p = IV] and q = [El. An even subgraph of G is a spanning subgraph of G in which each vertex has even degree. It is well known that the set of all even subgraph of G forms a vector space under the symmetric difference of subgraphs (where subgraphs are simply considered as subsets of E). We will denote this vector space by C(G) and consider it as a binary linear code. (The reader is referred to MacWilliams and Sloane (1977) and van Lint (1982) for background from coding theory.) Note that C(G) is a subspace of the vector space formed by all spanning subgraphs of G which is easily seen to be isomorphic to the q-dimensional vector space V(q, 2) of q-tuples with entries from GF(2). In this interpretation, we consider the coordinate positions to be indexed by the edges of G (in some fixed ordering); then each subgraph is associated with the corresponding (binary) characteristic vector of length q (which has an entry I in position e if and only ife belongs to the given subgraph). By abuse of notation, we will denote the subspace of V(q, 2) associated with all even subgraphs of G again by C(G). The vector space C(G) is (in either interpretation) usually called the cycle space of G; its dimension is known to be q - p + 1 provided that G is connected. It is clear that the minimum weight of a vector in C(G) is the smallest cardinality of a cycle in G, i.e. the girth 9 of G. We thus have the following well known result. Proposition 1. Let G be a connected graph with q edges on p vertices, and let 9 be the girth of G. Then C(G) is a binary [q, q - p + I, 9]-code. *) This note was written while the first author was visiting the Department of Combinatorics and Optimization of the University of Waterloo as an Adjunct Professor. He would like to thank his colleagues there for their hospitality. The second author acknowledges the support of the National Science and Engineering Research Counci! of Canada given under grant ~ 0GP0009258.