Our object is to explore "the s-tennis ball problem" (at each turn s balls are available and we play with one ball at a time). This is a natural generalization of the case s = 2 considered by Mallows and Shapiro. We show how this generalization is connected with s-ary trees and employ the notion of generating trees to obtain a solution expressed in terms of generating functions. Then, we presen...

Two graphs G1 and G2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (∆(G1)+1)(∆(G2)+1)≤ n+1, then G1 and G2 pack. Towards this conjecture, we show that for ∆(G1),∆(G2)≥ 300, if (∆(G1)+1)(∆(G2)+1)≤0.6n+1, then G1 and G2 pack. This is ...

13. K. Jordan. Chapters on the Classical Calculus of Probability. Akademiai Kiadb, Budapest, 1972. 14. P. A. MacMahon. Combinatory Analysis, Vols. I and II. New York: Chelsea, 1960. 15. G. P. Patil & J. K. Wani. "On Certain Structural Properties of the Logarithmic Series Distribution and the First Type Stirling Distribution." Sankhya, Series A, 27 (1965):271-180. 16. J. Riordan. An Introduction...

We prove that, for large n, every 3-connected D-regular graph on n vertices with D ≥ n/4 is Hamiltonian. This is best possible and verifies the only remaining case of a conjecture posed independently by Bollobás and Häggkvist in the 1970s. The proof builds on a structural decomposition result proved recently by the same authors.

Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan’s ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the rela...

A bisection of a graph G is a bipartition S1, S2 of V (G) such that −1 ≤ |S1|− |S2| ≤ 1. It is NP-hard to find a bisection S1, S2 of a graph G maximizing e(S1, S2) (respectively, minimizing max{e(S1), e(S2)}), where e(S1, S2) denotes the number of edges of G between S1 and S2, and e(Si) denotes the number of edges of G with both ends in Si. There has been algorithmic work on bisections, but ver...