The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to same labels. To date, this conjecture was mainly verified for complete and 3-colourable graphs. As strong support conjecture, it also proved admit such 4-labellings. In work, we invest...

In this work, we consider equitable proper labellings of graphs, which were recently introduced by Baudon, Pilśniak, Przybyło, Senhaji, Sopena, and Woźniak. Given a graph G, the goal is to assign labels edges so that (1) no two adjacent vertices are incident same sum labels, (2) every assigned about number times. Particularly, aim at designing such k-labellings G with k being as small possible....

Remark. Apropos of reduction mod p: If V is a Qp-vector space and G ⊂ GL(V ) is a compact subgroup, then there exists a G-fixed lattice in V for the following reason. Pick any lattice L ⊂ V . Then the G-stabilizer of L is open and of finite index. So Λ = ∑ g∈G gL ⊂ V is also a lattice, and it is definitely G-stable. The same works with coefficients in any finite extension of Qp, or even in Qp (...

We prove that the product version of 1-2-3 Conjecture, raised by Skowronek-Kaziów in 2012, is true. Namely, for every connected graph with order at least 3, we can assign labels 1, 2, 3 to edges such a way no two adjacent vertices are incident same labels.

We give a new proof of Kollár’s conjecture on the derived pushforward dualizing sheaf twisted by variation Hodge structure. This has been settled M. Saito via theory mixed modules and various applications in investigation Albanese maps. Our technique is $$L^2$$ theoretic which gives concrete constructions proofs to conjecture. The point view allows us generalize context non-abelian theory.