An algebraic characterization of 𝑘–colorability

Abstract

We characterize k-colorability of a simplicial graph via the intrinsic algebraic structure of the associated right-angled Artin group. As a consequence, we show that a certain problem about the existence of homomorphisms from right-angled Artin groups to products of free groups is NP-complete.

Group-based Cryptography in the Quantum Era

“…The previous approach permits in particular to convert graph theoretic problems for finite graphs into group theoretic ones for graph groups. Motivated by the fact that some of these group theoretic problems can be used for cryptographic purposes, such as authentication schemes, secret sharing schemes, zero-knowledge proofs, hash functions and key-exchange protocols, Flores, Kahrobaei, and Koberda have considered these groups as a promising platform for several cryptographic schemes (see [FKK19], [FKK21a], [FKK21b], [FKK22]). It is important, in this sense, that good knowledge of the group-theoretic structure of these groups (normal forms, centralizers, automorphisms, subgroups, etc.)…”

Path partial groups

“…Besides the aforementioned setup of quivers and their associated path algebras, the kind of study proposed in Problem 1.3 is well established in other fields as directed graphs and their associated Leavitt path algebras, see [1] or the Simplicity Theorem [1, 2.9.1] as a concrete instance of interaction between the graph side and its algebraic counterpart. In other areas as finite simplicial graphs and their associated right-angled Artin groups, studying this type of relations is also an active line of work, see [16,Problem 1.1]. In that work, the authors add the graph theoretical property of k-colorability to the list of properties of simplical graphs that can read off from the algebraic properties of its associated right-angled Artin group.…”