Equicovering matroids by distinct bases

AbstractWe ask, and answer, the following question: when is it possible to cover the elements of a given matroid M by p distinct bases so that each element is covered exactly k times? Our result is weaker than the Cyclic Order Conjecture of Wiedemann [9] (and of Kajitani, Ueno and Miyano [7]), but stronger than similar previous results; it could represent a first step towards resolving the conjecture

Green's function in partial subdivision networks

In the present work, we define a partial subdivision network of a given network G, by inserting a new vertex in some selected edges of G, so that each of these edges is replaced by two new edges with conductances that fulfil the Kirchhoff series law on the new network. Then, we obtain an expression for the Green kernel of the partial subdivision network in terms of the Green kernel of the base network. For that, we show the relation between Poisson problems on the partial subdivision network and Poisson problems on the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the partial subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a star network in which we have subdivided the even edges.Peer ReviewedPostprint (author's final draft

Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number

We investigate vector chromatic number, Lovasz theta of the complement, and quantum chromatic number from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e. that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the factors. We also prove an analog of Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value on the categorical product of graphs is equal to the minimum of its values on the factors. We conjecture that the analogous results hold for vector and quantum chromatic number, and we prove that this is the case for some special classes of graphs.Comment: 18 page

Green’s kernel of Schrödinger operators on generalized subdivision networks

Peer ReviewedPostprint (author's final draft