We conjecture that every nite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1). We show that it suuces to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for nite simple groups, such a short presentation is computable in polynomia...

‎An  oriented perfect path double cover (OPPDC) of a‎ ‎graph $G$ is a collection of directed paths in the symmetric‎ ‎orientation $G_s$ of‎ ‎$G$ such that‎ ‎each arc‎ ‎of $G_s$ lies in exactly one of the paths and each‎ ‎vertex of $G$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete‎ ‎Math‎. ‎276 (2004) 287-294) conjectured that ...

Let G be any abelian group and {asGs}ks=1 be a finite system of cosets of subgroups G1, . . . , Gk. We show that if {asGs} k s=1 covers all the elements of G at least m times with the coset atGt irredundant then [G : Gt] 6 2 and furthermore k > m + f([G : Gt]), where f( ∏ r i=1 p αi i ) = ∑ r i=1 αi(pi − 1) if p1, . . . , pr are distinct primes and α1, . . . , αr are nonnegative integers. This ...

In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and ∆(G) ≤ r, which has the most complete subgraphs of size t, for t ≥ 3. The conjectured extremal graph is aKr+1 ∪Kb, where n = a(r + 1) + b with 0 ≤ b ≤ r. Gan, Loh, and Sudakov proved the conjecture wh...