Let \(T_{g}(m,n)\) (respectively, \(P_{g}(m, n)\)) be the number of rooted maps, on an orientable (respectively, non-orientable) surface of type \(g\), which have \(m\) vertices and \(n\) faces. Bender, Canfield and Richmond [3] obtained asymptotic formulas for \(T_{g}(m,n)\) and \(P_{g}(m,n)\) when \(\epsilon \leq m/n \leq 1/\epsilon\) and \(m,n \to \infty\). Their formulas cannot be extended to the extreme case when \(m\) or \(n\) is fixed. In this paper, we shall derive asymptotic formulas for \(T_{g}(m,n)\) and \(P_{g}(m,n)\) when \(m\) is fixed and derive the distribution for the root face valency. We also show that their generating functions are algebraic functions of a certain form. By the duality, the above results also hold for maps with a fixed number of faces.

Consider the following two-person game on the graph \(G\). Player I and II move alternatingly. Each move consists in coloring a yet uncolored vertex of \(G\) properly using a prespecified set of colors. The game ends when some player can no longer move. Player I wins if all of \(G\) is colored. Otherwise Player II wins. What is the minimal number \(\gamma(G)\) of colors such that Player I has a winning strategy? Improving a result of Bodlaender [1990] we show \(\gamma(T) \leq 4\) for each tree \(T\). We, furthermore, prove \(\gamma(G) = O(\log |G|)\) for graphs \(G\) that are unions of \(k\) trees. Thus, in particular, \(\gamma(G) = O(\log |G|)\) for the class of planar graphs. Finally we bound \(4(G)\) by \(3w(G) – 2\) for interval graphs \(G\). The order of magnitude of \(\gamma(G)\) can generally not be improved for \(k\)-fold trees. The problem remains open for planar graphs.

We examine properties of a class of hypertrees, occurring in probability, which are described by sequences of subscripts.

We give, among other results, a new method to construct for each positive integer \(n\) a class of orthogonal designs \( {OD}(4^{n+1};m;4^n m,4^n m,4^n m,4^n m)\), \(m=2^a 10^b 26^c +4^n+1\), \(a,b,c\) non-negative integers.

We verify that \(6\) more of the tum squares of order \(10\) cannot be completed to a triple of mutually orthogonal Latin squares of order \(10\). We find a pair of orthogonal Latin squares of order \(10\) with \(6\) common transversals, \(5\) of which have only a single intersection, and a pair with \(7\) common transversals.

We give a complete solution to the existence problem for subdesigns in complementary \(\mathrm{P}_3\)-decompositions, where \(\mathrm{P}_3\) denotes the path of length three. As a corollary we obtain the spectrum for incomplete designs with block size four and \(\lambda = 2\), having one hole.

In this paper, we give some recursive constructions for large sets of disjoint group-divisible designs with block size \(3\). In particular, we construct new infinite classes of large sets for designs having group-size two. These large sets have applications in cryptography to the construction of perfect threshold schemes.

A decomposition into non-isomorphic matchings, or \(DINIM\) for short, is a partition of the edges of a graph \(G\) into matchings of different sizes. As a special case of our results, we prove that if a graph \(G\) has at least \((2\chi’ – 2)\chi’ + 1\) edges, where \(\chi’ = \chi'(G)\) is the chromatic index of \(G\), then \(G\) has a \(DINIM\). In particular, the \(n\)-dimensional cube, \(Q_n\), \(n \geq 4\), has a \(DINIM\). These results confirm two conjectures which appeared in Chinn and Richter [3].