Architettura come Ars Combinatoria = Architecture as Ars Combinatoria

L\u2019architettura non rappresenta pi\uf9, oggi, il limite della sperimentazione e dell\u2019innovazione. E\u2019 un\u2019arte estremamente low-tech il cui campo di possibilit\ue0 \ue8 quello di scegliere e usare quello che gi\ue0 \ue8 stato fatto in passato. E\u2019 per questo che la principale abilit\ue0 di un progettista dovrebbe essere quella di ri-conoscere le forme della complessit\ue0 all\u2019intorno: sono le citt\ue0 ad essere il pi\uf9 ampio palinsesto delle possibilit\ue0 dell\u2019architettura, nel rapporto biunivoco che si instaura tra una citt\ue0 che prende forma da un\u2019architettura e un\u2019architettura che ha le sue ragioni nella coeva citt\ue0. L\u2019architettura ha cos\uec il suo fondamento nell\u2019ospitalit\ue0, elemento che la distingue nettamente dal design

Chordal Graphs are Fully Orientable

Suppose that D is an acyclic orientation of a graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let m and M denote the minimum and the maximum of the number of dependent arcs over all acyclic orientations of G. We call G fully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is called chordal if every cycle in G of length at least four has a chord. We show that all chordal graphs are fully orientable.Comment: 11 pages, 1 figure, accepted by Ars Combinatoria (March 26, 2010

On the tensor degree of finite groups

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x \otimes y= 1_{_{G \otimes G}}$ in the nonabelian tensor square $G \otimes G$ of $G$. This number, divided by $|G|^2$, is called the tensor degree of $G$ and has connection with the exterior degree, introduced few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335--343]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.Comment: 10 pages, accepted in Ars Combinatoria with revision

Full Orientability of the Square of a Cycle

Let D be an acyclic orientation of a simple graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define m and M to be the minimum and the maximum number of d(D) over all acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n is fully orientable except n=6.Comment: 7 pages, accepted by Ars Combinatoria on May 26, 201