# Complex

A complex is an edge-colored graph that generalizes the graph encoding of a map and the flag graph of a polytope.

## Definition

There exist several equivalent ways to define the idea of a complex.

### Edge colored graph

A complex is a connected properly edge colored simple graph.

To put it another way, a complex is a vertex set V, a color set C, and a function E from colors where $A(i)$ yields the edges with color i such that:

• For every color i the every vertex is incident on exactly 1 edge of color i.
• If $i \neq j$ then the intersection $A(i)\cup A(j)$ is empty.
• The graph $(V,A)$ is connected.

### Permutation sequence

An n-complex is a set $\Omega$ and a sequence of permutations on $\Omega$ , $A : \{1,\dots,n\} \rightarrow \mathrm{Aut}(\Omega)$ such that:

• For every $A(i)$ in A, $A(i)\circ A(i)=\mathrm{id}$ • For every $A(i)$ in A, there is no element of $\Omega$ which is a fixed point of $A(i)$ .
• If $i \neq j$ then for any element $x\in\Omega$ , $A(i)(x)\neq A(j)(x)$ .
• The group generated by the permutations in A is transitive on $\Omega$ .

## Concepts

### Elements The edges, faces and vertices of tetrahedron flag graph shown above.

For a given i the elements of a complex of rank i correspond to the orbits of the group generated by permutations $A(j)$ where $i\neq j$ .

Alternatively the elements of rank i in a a complex correspond to the connected components in the graph with edges of color i removed.

### Incidence

Two elements are incident on each other if their connected components share a vertex.