The well-known 1–2–3 Conjecture asserts that the edges of every graph without isolated can be weighted with 1, 2 and 3 so adjacent vertices receive distinct degrees . This is open in general, while it known to possible from weight set { 1 , 4 5 } We show for regular graphs sufficient use weights 2, 3, 4. Moreover, we prove conjecture hold d -regular ≥ 10 8

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin’s Conjecture, which we call geometric set. We construct del Pezzo surface degree 1 whose is Zariski dense. In particular, this provides first counterexample to original version Conjecture dimension 2 characteristic 0. Assuming finiteness Tate-Shafarevich groups elliptic curves over $${\...

Olivier Baudon y, Guillaume Fertin y and Ivan Havel z y LaBRI U.M.R. CNRS 5800, Universit e Bordeaux I 351 Cours de la Lib eration, F33405 Talence Cedex z Faculty of Mathematics and Physics, Charles University Malostransk e n am. 25, 118 00 Praha, Czech Republic fbaudon,[email protected], [email protected] Abstract We study an n-dimensional directed symmetric hypercube Hn, in which e...

We begin by giving some background to this result. A matroid M is an excluded minor for a minor-closed class of matroids if M is not in the class but all proper minors of M are. It is natural to attempt to characterize a minor-closed class of matroids by giving a complete list of its excluded minors. Unfortunately, a minor-closed class of matroids can have an infinite number of excluded minors....