On Compact Symmetric Regularizations of Graphs

Let G be a finite simple graph of order n, maximum degree Δ, and minimum degree δ. A compact regularization of G is a Δ-regular graph H of which G is an induced subgraph: H is symmetric if every automorphism of G can be extended to an automorphism of H. The index |H:G| of a regularization H of G is the ratio |V(H)|/|V(G)|. Let mcr(G) denote the index of a minimum compact regularization of G and let mcsr(G) denote the index of a minimum compact symmetric regularization of G. Erdős and Kelly proved that every graph G has a compact regularization and mcr(G)≤2. Building on a result of König, Chartrand and Lesniak showed that every graph has a compact symmetric regularization and mcsr(G)≤2Δ−δ. Using a partial Cartesian product construction, we improve this to mcsr(G)≤Δ−δ+2 and give examples to show this bound cannot be reduced below Δ−δ+1

ISM enrichment and local pollution in dwarf galaxies

The fate of metals after they are released in starburst episodes is still unclear. What phases of the interstellar medium are involved, in which timescales? Evidence has grown over the past few years that the neutral phase of blue compact dwarf (BCD) galaxies may be metal- deficient as compared to the ionized gas of their HII regions. These results have strong implications for our understanding of the chemical evolution of galaxies. We review here the main results and the main caveats in the abundance determination from far-UV absorption-lines. We also discuss possible scenarios concerning the journey of metals into the interstellar medium, or even their ejection from the galaxy into the intergalactic medium.Comment: Long version of a proceeding for the conference "A Universe of Dwarf Galaxies: Observations, Theories, Simulations" held in Lyon, France (14th-18th, June 2010