In a seminal paper [6], Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of G obtained via their method, was later coined the Alon-Tarsi number of G and was denoted by AT (G) (see e.g. [24]). They have provided a combinatorial inter...

The main theme of this article is that counting orbits of an in nite permutation group on nite subsets or tuples is very closely related to combinatorial enumeration this point of view ties together various dis parate stories Two graphs and even graphs The rst story originated with Neil Sloane when he was compiling the rst edition of his dictionary of integer sequences He observed that certain ...

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for cer...

PairViz is an R package that produces orderings of statistical objects for visualization purposes. We abstract the ordering problem to one of constructing edge-traversals of (possibly weighted) graphs. PairViz implements various edge traversal algorithms which are based on Eulerian tours and Hamiltonian decompositions. We describe these algorithms, their PairViz implementation and discuss their...

De Bruijn graphs provide a framework for genome assembly, where the correct reconstruction of the genome is given by one of the many Eulerian tours through the graph. The assembly problem is complicated by genomic repeats, which allow for many possible Eulerian tours, thereby increasing the de Bruijn graph complexity. Optical maps provide an ordered listing of restriction fragment sizes for a g...

Nash-Williams’ well-balanced orientation theorem [11] is extended for orienting several graphs simultaneously. We prove that if G1, ..., Gk are pairwise edge-disjoint subgraphs of a graph G, then G has a well-balanced orientation ~ G such that the inherited orientations ~ Gi of Gi are well-balanced for all 1 ≤ i ≤ k. We also have new results about simultaneous well-balanced orientations of non-...

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for cer...