The chromatic numbers of double coverings of a graph

If we fix a spanning subgraph $H$ of a graph $G$, we can define a chromatic number of $H$ with respect to $G$ and we show that it coincides with the chromatic number of a double covering of $G$ with co-support $H$. We also find a few estimations for the chromatic numbers of $H$ with respect to $G$.Comment: 10 page

On Stable embeddability of partitions

Several natural partial orders on integral partitions, such as the embeddability, the stable embeddability, the bulk embeddability and the supermajorization, raise in the quantum computation, bin-packing and matrix analysis. We find the implications between these partial orders. For integral partitions whose entries are all powers of a fixed number $p$, we show that the embeddability is completely determined by the supermajorization order and we find an algorithm to determine the stable embeddability.Comment: 10 page

Enumerations of finite topologies associated with a finite graph

The number of topologies and non-homeomorphic topologies on a fixed finite set are now known up to $n=18$, $n=16$ but still no complete formula yet (Sloane). There are one to one correspondence among topologies, preorder and digraphs. In this article, we enumerate topologies and non-homeomorphic topologies whose underlying graph is a given finite graph.Comment: 2 figure

Enumeration of the BranchedmZp-Coverings of Closed Surfaces

AbstractIn this paper, we enumerate the equivalence classes of regular branched coverings of surfaces whose covering transformation groups are the direct sum of m copies of Zp, p prime