Transforming One Plane Graph Into Another

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

If G is a plane graph with nice properties (e.g. k-valent, all faces k-gons, has an Eulerian circuit, has no Eulerian circuit) then it is of interest to construct related graphs which either preserve these properties or show the new graphs lack these properties.

Examples of ways to transform one plane graph into another graph (perhaps not plane) are:

a. Line graph

b. Dual graph

c. Medial graph

d. Truncation graph

f. Replace an edge with a hexagon

Questions which one can ask are:

i. Does the operation preserve the graph having an Eulerian circuit?

ii. Does the operation preserve the graph having a Hamiltonian circuit?

iii. If the original graph had a cut-through Eulerian circuit, will the transformed graph have this property?

iv. If the original graph has an Eulerian circuit which cuts through at no vertex, will the transformed graph have this property?

v. If the original graph is pancyclic will the new graph be pancyclic?

Infinite families of graphs which have or fail to have a property are more noteworthy than isolated examples.

Note:

The questions above are framed for plane graphs in general, however, within the plane graphs, the following graphs are of especial interest:

a. 3-polytopal graphs

b. k-valent, 3-polytopal graphs (k = 3, k = 4, k = 5)

c. k-gon, 3-polytopal graphs (k = 3, k = 4, k = 5) (Plane graphs all of who faces are triangles are sometimes called plane triangulations.)

Definitions:

A (simple) graph is 3-connected if for any pair of vertices u and v in the graph there are three path from u to v which have only the vertices u and v in common.

A (simple) graph is pancyclic if it has circuits of all lengths from 3 to n, where n is the number of vertices of the graph.