In this paper we give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian graph of bounded treewidth. The problems of counting ETs are known to be ♯P complete for general graphs (Brightwell and Winkler, 2005 [4]). To date, no polynomial-time algorithm for counting Euler tours of any class of graphs is known except for the very special case of seri...

Isotropic systems are structures which unify some properties of 4-regular graphs and of pairs of dual binary matroids. In this paper we unify the properties of the symmetric Tutte polynomials (i.e. with equal variables) of binary matroids and of the Martin polynomials of 4-regular graphs. For this purpose we introduce the orienting vectors of an isotropic system in order to generalize the euler...

Circulant graphs are an extremely well-studied subclass of regular graphs, partially because they model many practical computer network topologies. It has long been known that the number of spanning trees in n-node circulant graphs with constant jumps satisfies a recurrence relation in n. For the non-constant-jump case, i.e., where some jump sizes can be functions of the graph size, only a few ...

The agroforestry literature makes references to HAHA designs. This paper is the first systematic study of these designs and some of their generalizations. The relationship between HAHA designs and Eulerian paths around bipartite graphs is exploited.