INCIDENCE GEOMETRIES

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INCIDENCE GEOMETRIES

• Motivation and definitions

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Plan

• In this chapter we will cover the following
• motivation
• incidence geometries
• incidence structures
• combinatorial configurations

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Incidence structure

• An incidence structure C is a triple
• C (P,L,I) where
• P is the set of points,
• L is the set of blocks or lines
• I ? P ? L is an incidence relation.
• Elements from I are called flags.
• The bipartite incidence graph G(C) with black
  vertices P, white vertices L and edges I is
  known as the Levi graph of the structure C.

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Examples

• 1. Each graph G (V,E) is an incidence
  structure P V, L E, (x,e) 2 I if and only if
  x is an endvertex of e.
• 2. Any family of sets F µ P(X) is an incidence
  structure. P X, L F, I 2.
• 3. A line arrangement L l1, l2, ..., ln
  consisting of a finite number of n distinct lines
  in Euclidean plane E2 defines an incidence
  structure. Let V denote the set of points from E2
  that are contained in at least two lines from L.
  Then P V, L L and I is the point-line
  incidence in E2.

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Exercises

• N1. Draw the Levi graph of the incidence
  structure defined by the complete bipartite graph
  K3,3.
• N2. Draw the Levi graph of the incidence
  structure defined by the powerset P(a,b,c).
• N3. Determine the Levi graph of the incidence
  structure, defined by an arrangemnet of three
  lines forming a triangle in E2.

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Incidence geometry

• Incidence geometry (G,c)of rank k is a graph G
  with a proper vertrex coloring c, where k colors
  are used.
• Sometimes we denote the geometry by (G,c,I,).
  Here cVG ! I is the coloring and I k is the
  number of colors, also known as the rank of G.
  Also is the incidence.
• I is the set of types. Note that only object of
  different types may be incident.

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Examples

• 1. Each incidence structure is a rank 2 geometry.
  (Actualy, look at its Levi graph.)
• 2. Each 3 dimensional polyhedron is a rank 3
  geometry. There are three types of objects
  vertices, edges and faces with obvious geometric
  incidence.
• 3. Each (abstract) simplicial complex is an
  incidence geometry.
• 4. Any complete multipartite graph is a geometry.
  Take for instance K2,2,2, K2,2,2,2, K2,2, ..., 2.

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Pasini Geometry

• Pasini defines incidence geometry (that we call
  Pasini geometry) in more restrictive way.
• For k1, the graph must contain at least two
  vertices V(G)gt1.
• For kgt1
• G has to be connected,
• For each x ? V(G) the (k-1)-colored graph
  (Gx,c), called residuum, induced on the neigbors
  of x is a Pasini geometry of rank (k-1).

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Incidence geometries of rank 2

• Incidence geometries of rank 2 are simply
  bipartite graphs with a given black and white
  vertex coloring.
• Rank 2 Pasini geometries are in addition
  connected and the valence of each vertex is at
  least 2 d(G) gt1.

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Example of Rank 2 Geometry

• Graph H on the left is known as the Heawood
  graph.
• H is connected
• H is trivalent d(H) D(H) 3.
• H je bipartite.
• H is a Pasini geometry.

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Another View

• Geometry of the Heawood graph H has another
  interpretation.
• Rank 2. There are two types of objects in
  Euclidean plane, say, points and curves.
• There are 7 points, 7 curves, 3 points on a
  curve, 3 curves through a point.
• The corresponding Levi graph is H!

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In other words ...

• The Heawood graph (with a given black and white
  coloring) is the same thing as the Fano plane
  (73), the smallest finite projective plane.
• Any incidence geometry can be interpeted in terms
  of abstract points, lines.
• If we want to distinguish geometry
  (interpretation) from the associated graph we
  refer to the latter the Levi graph of the
  corresponding geometry.

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Simplest Rank 2 Pasini Geometries
Cycle (Levi Graph)

• Simplest geometries of rank 2 in the sense of
  Pasini are even cycles. For instance the Levi
  graph C6 corresponds to the triangle.

Triangle (Geometry)
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Rank 3

• Incidence geometries of rank 3 are exactly
  3-colored graphs.
• Pasini geometries of rank 3 are much more
  restricted. Currently we are interested in those
  geometries whose residua are even cycles.
• Such geometries correspond to Eulerian surface
  triangulations with a given 3-vertex coloring.

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Flag System as Geometries

• Any flag system ? µ V E F defines a rank 3
  geometry on X V t E t F. There are three types
  of elements and two distinct elements of X are
  incicent if and only if they belong to the same
  flag of ?.

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Exercises

• N1. Prove that the Petrie dual of a self-avoiding
  map is self-avoiding.
• N2. Prove that any operation Du,Tr,Me,Su1, ... of
  a self-avoiding map is self-avoiding.
• N3. Prove that BS of any map is self-avoiding.
• N4. Show that any self-avoding map may be
  considered as a geometry of rank 4 (add the
  fourth involution).

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Self-avoiding maps as Geometries of rank 4

• Consider a generalized flag system ? µ V E F
  P that defines a rank 4 geometry on X V t E t
  F t P.
• There are four types of elements and two distinct
  elements of X are incident if and only if they
  belong to the same flag of ?.
• We may take any self-avoiding map M and the four
  involutions ?0,?1,?2 and ?3 and define the above
  geometry.

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Homework

• H1 Describe the rank 4 geometry of the projective
  planar map on the left.

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Geometries from Groups

• Let G be a group and let G1,G2,...,Gk be a
  family of subgroups of G.
• Form the cosets xGt, t 2 1,2, ..., k.
• An incidence geometry of rank k is obtained as
  follows
• Elements of type t 2 1,2,...,k are the cosets
  xGt.
• Two cosets are incident xGt yGs if and only if
  xGt Å yGs ¹ .

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Q The Quaternion Units
Q 1 -1 i -i j -j k -k
1 1 -1 i -i j -j k -k
-1 -1 1 -i i -j j -k k
i i -i -1 1 k -k -j j
-i -i i 1 -1 -k k j -j
j j -j k -k -1 1 i -i
-j -j j -k k 1 -1 -i i
k k -k j -j -i i -1 1
-k k -k j -j -i i 1 -1
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Geometry from Quaternions

• Example Q 1,-1,i,-i,j,-j,k,-k.
• Gi 1,-1,i,-i, Gj 1,-1,j,-j, Gk
  1,-1,k,-k.

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Quaternions - Continiuation
j,k

• Levi graph is an octahedron.
• Labels on the left
• i 1,-1,i,-i
• j,k j,-j,k,-k, etc.

j
k
i
i,j
i,k
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Quaternions Examle of Rank 4 Geometry.
j,k

• Levi graph was an octahedron.
• Notation
• i 1,-1,i,-i
• j,k j,-j,k,-k, itd.
• If we add the sugroup G0 1,-1, four more
  cosets are obtained
• Additional notation
• 1 1,-1,ii,-i, etc.

k
j
k
1
j
i
i
i,k
i,j