3

Let $H=\left( E_0 ,E_1 ,E_2 , \ldots , E_d \right)$ be a $d$-dimensional full-hyper graph/complex. That is to say, if for some $i\in \left[d \right]$ the hyper-edge $e_j \in E_i$ than for any $i-1$-dimensional $e_k \subset e_j$: $e_k \in E_{i-1}$. The vertices are $E_0$, the hyper-edges between $2$ vertices are $E_1$, between $3$ are $E_2$ and so on.

I bring forward 2 definitions of regularity, and I'd like to show their equivalence:

  • There exists $r_1,\ldots , r_d$ such that for each $1\leq i \leq d$ each vertex $v\in E_0$ has exactly $r_i$ hyper-edges from $E_i$ to which it belongs.
  • There exists $r_1,\ldots , r_d$ such that for each $1\leq i \leq d$ each $e_j \in E_{i-1}$ has exactly $r_i$ hyper-edges from $E_i$ that contain it.

I claim that the definitions are equivalent, obviously with different values of $r_i$'s.

My attempted proof for the case of $d=3$:

$\Leftarrow$

Each vertex $v$ has $r_1$ $2$-edges, each edge has $r_2$ $3$-edges. Each $3$-edge, by definition, uses exactly a distinct pair of $v$'s $2$-edges. Therefore, each $3$-edge is counted twice - one per each $2$-edge of $v$. This makes the total number of $3$-edges exactly $\frac{r_1\cdot r_2}{2}$.

Having a bit trouble for the other direction. Could not disprove it, and I believe it holds, but not every pair of edges that share a common vertex must yield a $3$-edge.

Also, I'd like to try to generalize this for any $d$, but first, to understand the other direction for $d=3$.

Are the 2 definitions for regularity equivalent? I have 2 questions:

  • For the case of d=3?
  • For any d?

(What I provided was a proof in one direction. As for the other direction, I am not sure it holds.)

John L.
  • 39,205
  • 4
  • 34
  • 93
Benicio Agüero
  • 142
  • 10

1 Answers1

3

Consider the following full hypergraph $H$ (or abstract simplicial complex) of dimension $2$, represented as a simplicial complex (nodes are in black, $2$-edges in red, $3$-edges are turqoise triangles): enter image description here

(or, algebraically: $E_0 = \{v_1,v_2,v_3,v_4,v_5, v_6\}$, $E_1=\{(v_1,v_2), (v_2,v_3), (v_3,v_4), (v_4,v_5), (v_5,v_6), (v_1,v_6), (v_1,v_4), (v_2,v_6), (v_3,v_5)\}$, $E_2 =\{(v_1,v_2,v_6),(v_3,v_4,v_5)\}$ )

$H$ is regular according to the first definition, since each vertex belongs to 3 $2$-edges and 1 triangle. However, $H$ is not regular according tot the second definition, because e.g. $(v_1,v_2)$ is contained in 1 triangle while $(v_2,v_3)$ is contained in 0 triangles.

This means the "other direction" does not hold for dimensions $2$ and higher, so the two properties are not equivalent. (Of course, the two definitions are trivially equivalent for dimension $1$)

As for whether the second property implies the first, your proof (which seems to be for dimension $2$) looks correct to me. The step of going from $2$-edges to $3$-edges can be generalized into $i$-edges to $(i+1)$-edges, after the proof follows by induction. So, it seems the second property is stronger than the first.

Discrete lizard
  • 8,392
  • 3
  • 25
  • 53