This paper presents a theorem concerning a matroid with the parity condition. The theorem provides matroid generalizations of graph-theoretic results of Lewin [3] and Gallai [1]. Let M=(E, F) be a matroid, where E is a finite set of elements and F is the family of independent sets of M (in this paper, we presuppose a knowledge of matroid theory; our standard reference is Welsh [4]). Assume that...

Autonomous programmable computing devices made of biomolecules could interact with a biological environment and be used in future biological and medical applications. Biomolecular implementations of finite automata and logic gates have already been developed. Here, we report an autonomous programmable molecular system based on the manipulation of DNA strands that is capable of performing simple...

In this paper we introduce the notion of tolerance in connection with Helly type theorems and prove, using the Erdős-Gallai theorem, that any Helly type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number of exceptional sets or points. In particular, we analyze some of the classical Helly type theorems, such as Caratheodory’s and Tverberg’s theorems,...

We say that a graph G has the CIS-property and call it a CIS-graph if every maximal clique and every maximal stable set of G intersect. By definition, G is a CIS-graph if and only if the complementary graph G is a CISgraph. Let us substitute a vertex v of a graph G′ by a graph G′′ and denote the obtained graph by G. It is also easy to see that G is a CIS-graph if and only if both G′ and G′′ are...

We discuss certain extremal problems in combinatorial geometry, including Sylvester’s problem and its generalizations.

The graph realization problem is to find for given nonnegative integers a1, . . . , an a simple graph (no loops or multiple edges) such that each vertex vi has degree ai. Given pairs of nonnegative integers (a1, b1), . . . , (an, bn), (i) the bipartite realization problem ask whether there is a bipartite graph (no loops or multiple edges) such that vectors (a1, ..., an) and (b1, ..., bn) corres...

We formulate and study a new variant of the k-armed bandit problem, motivated by e-commerce applications. In our model, arms have (stochastic) lifetime after which they expire. In this setting an algorithm needs to continuously explore new arms, in contrast to the standard k-armed bandit model in which arms are available indefinitely and exploration is reduced once an optimal arm is identified ...