Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] v...

Two variations of set intersection representation are investigated and upper and lower bounds on the minimum number of labels with which a graph may be represented are found which hold for almost all graphs. Speciically, if k (G) is deened to be the minimum number of labels with which G may be represented using the rule that two vertices are adjacent if and only if they share at least k labels,...

Fermat's Last Theorem—which we shall abbreviate to FLT—is the (as yet unproved) assertion that the Diophantine equation x" + y" — z" has no solutions in positive integers if 11 ^ 3. It would suffice to deal with the case in which n is prime, and this is where the most significant work has been done. None the less it is not yet known even whether FLT is true for infinitely many prime exponents. ...

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0, 1, ..., N}, chosen randomly according to a binomial model with parameter p(N), with N = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O’Bryant. Furthermore, we ex...

One potential method to attain more energy-efficient circuits with the current technology is Near-Threshold Computing, which means using less energy per gate by designing the supply voltages to be closer to the threshold voltage of transistors. However, this energy savings comes at a cost of a greater probability of gate failure, which necessitates that the circuits must be more fault-tolerant,...

We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ¬ACω in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular c...

A symmetric m×m matrix M with entries taken from {0, 1, ∗} gives rise to a graph partition problem, asking whether a graph can be partitioned into m vertex sets matched to the rows (and corresponding columns) of M such that, if Mij = 1, then any two vertices between the corresponding vertex sets are joined by an edge, and if Mij = 0 then any two vertices between the corresponding vertex sets ar...

Let G be a finite cyclic group with generator α and with an encoding so that multiplication is computable in polynomial time. We study the security of bits of the discrete log x when given expα(x), assuming that the exponentiation function expα(x) = α is oneway. We reduce he general problem to the case that G has odd order q. If G has odd order q the security of the least-significant bits of x ...