Euler Graph

The term "Euler graph" is sometimes used to denote a graph for which all vertices are of even degree (e.g., Seshu and Reed 1961). Note that this definition is different from that of an Eulerian graph, though the two are sometimes used interchangeably and are the same for connected graphs.

The numbers of Euler graphs with , 2, ... nodes are 1, 1, 2, 3, 7, 16, 54, 243, 243, 2038, ... (OEIS A002854; Robinson 1969; Mallows and Sloane 1975; Buekenhout 1995, p. 881; Colbourn and Dinitz 1996, p. 687), the first few of which are illustrated above. There is an explicit formula giving these numbers.

There are more Euler graphs than Eulerian graphs since there exist disconnected graphs having multiple disjoint cycles with each node even but for which no single cycle passes through all edges. The numbers of Euler-but-not-Eulerian graphs on , 2, ... nodes are 0, 0, 0, 0, 0, 1, 2, 7, 20, 76, 334, 2498, ... (OEIS A189771), the first few of which are illustrated above.

REFERENCES

Buekenhout, F. (Ed.). Handbook of Incidence Geometry: Building and Foundations. Amsterdam, Netherlands: North-Holland, 1995.

Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, 1996.

Mallows, C. L. and Sloane, N. J. A. "Two-Graphs, Switching Classes, and Euler Graphs are Equal in Number." SIAM J. Appl. Math. 28, 876-880, 1975.

Robinson, R. W. "Enumeration of Euler Graphs." In Proof Techniques in Graph Theory (Ed. F. Harary). New York: Academic Press, pp. 147-153, 1969.

Seshu, S. and Reed, M. B. Linear Graphs and Electrical Networks. Reading, MA: Addison-Wesley, 1961.Sloane, N. J. A. Sequences A002854/M0846 and A189771 in "The On-Line Encyclopedia of Integer Sequences."