Enumeration of Eulerian and unicursal planar maps
By a unicursal map the authors mean a 2-cell embedding in the sphere of a graph (loops and multiple edges allowed) that has exactly two vertices of odd degree. Let \(U'(n)\) denote the number of rooted unicursal maps with \(n\) edges and let \(U_i'(n)\) the number of rooted unicursal maps with \(n\) edges and \(i\) vertices of degree 1 (clearly \(i = 0\), \(1\) or \(2\)). Among several results, the authors prove: Theorem 1. \(U'(n)= 2^{n-2}{2n\choose n}\), \(n\geq 1\), and for \(n\geq 2\), \(U_0'(n)= 2^{n-2} {n-2\over n} {2n-2\choose n-1}\), \(U_1'(n)= 2^{n-1}{2n-2\choose n-1}\), and \(U_2'(n)= 2^{n-2}{2n- 2\choose n-1}\).
scientific article; zbMATH DE number 2083697
scientific article