Kempe chain

Kempe chain

Alfred B. Kempe
(

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Kempe.html

bio at St Andrews)
first used these chains, now called after him,
in 1879 in a “proof” of the

four-color conjecture

. Although Percy Heawood
found a flaw in his proof 11 years later, the idea of

Kempe chains

itself is
quite sound. Heawood used it to prove 5 colors suffice for maps on the plane,
and the 1976 proof by Appel, Haken and Koch is also based on Kempe’s ideas.

The original Kempe chains were used in the

context

of

colorings

of countries
on a map, or in modern terminology

face colorings

of a

plane graph

G

(such that no two

adjacent

faces receive the same color). The idea was
extended by Heawood to

embeddings

of a graph in any other surface. Here

•

a Kempe chain of colors

a

and

b

is a maximal connected set of faces
that have either of those colors. as in: you can travel from any face in the set to any other, through the set. Maximal as in: there are no more faces of those colors you could enlarge the set with, i.e. that border the area you’ve already got.

In the modern dual formulation of the

four-color theorem

, faces of

G

are
replaced by vertices of the

dual graph

G

*

(and vice versa); vertices are
adjacent (linked by an edge) in

G

*

whenever the corresponding faces of

G

were adjacent (sharing a border). It now becomes a

vertex coloring

problem (again,

adjacent vertices

must receive different colors). Here

•

a Kempe chain of colors

a

and

b

is a maximal connected subgraph
containing only vertices of those colors.

Connected

as usual in

graph theory

(there’s a path between any two vertices) and maximal as in: there are no
more vertices of those colors you could enlarge the

subgraph

with, i.e. that
are adjacent to a vertex you’ve already got.

An alternative formulation: let

G

⁢

(

a

,

b

)

be the subgraph induced by all the
vertices of color

a

or

b

(that is, with all the edges that run between
those vertices). Now any

connected component

H

⁢

(

a

,

b

)

of

G

⁢

(

a

,

b

)

is a Kempe
chain.

While faces imply an embedding in a surface, the vertex version of the

definition

does not rely on any embedding. The chains are more useful in
the context of an embedding though.

Kempe chains of edges

With

edge coloring

of graphs (again, with different colors for adjacent
edges i.e. those that meet at a vertex), an analogous concept can defined.
Here

•

a Kempe chain of colors

a

and

b

is a maximal connected subgraph
where the edges have either of those colors. Connected and maximal as before.

An alternative formulation: let

G

⁢

(

a

,

b

)

be the subgraph consisting of all the
edges of color

a

or

b

(with any vertices

incident

to them). Now any
connected component

H

⁢

(

a

,

b

)

of

G

⁢

(

a

,

b

)

is a Kempe chain.

While superficially

similar

to the previous definition, these “Kempe chains”
are rather different animals. Their

structure

is far simpler. At any vertex of

H

⁢

(

a

,

b

)

, there can be at most one edge of color

a

and one edge of color

b

(at most two edges in all). Two things can happen:

∘

Every vertex in the chain has two such edges. In a

finite
graph

, this must mean a

closed path

(cycle). Note that, being colored
alternatingly, the number of edges must now be even.

∘

A vertex misses out on having an edge of one of those colors,
and the chain stops there (it can’t miss out on both colors because then
it wouldn’t be part of the chain). In a finite graph, the other end of
the chain must now also terminate somewhere (at a vertex that misses
out either one of the colors). The chain is an open path of one or
more edges (its length can be even or odd).

Kempe chain arguments

One technique that can be used with all these types of chains is

•

swapping the colors in one

H

⁢

(

a

,

b

)

. This is always possible:
by definition, there are no adjacent items that could
lead to a color clash.

Kempe slipped up when he swapped an

H

⁢

(

a

,

b

)

and an

H

⁢

(

a

,

c

)

without
taking in account that swapping colors

a

and

b

in part of the
graph could alter the shape and connectedness of any

H

⁢

(

a

,

c

)

, but
swapping colors in one Kempe chain at a time (and then taking stock of
the lay of the land afresh) is quite safe.

Swapping can be used to free up a color somewhere, see the

http://planetmath.org/node/6932

proof of Vizing’s

theorem

for a repeated use
of this ploy on edge-based Kempe chains.

Specific to the use in plane graphs we have that

•

if a Kempe chain forms a cycle, it disconnects the sphere or plane area
into

disjoint

parts (on the plane, inside and outside the cycle).

Specifically when four colors are used for faces:

•

The whole area is divided into red/green swathes and yellow/blue ones.
Alternatively, into red/yellow and green/blue ones. Or red/blue and
yellow/green ones.

Specifically for edge-based Kempe chains in

regular

ρ

-valent graphs, if
we try to edge-color the graph with only

ρ

colors:

•

No vertex can miss out on any of the

ρ

colors, so every

H

⁢

(

a

,

b

)

must be a cycle (of even length).

Title

Kempe chain

Canonical

name

KempeChain

Date of creation

2013-05-16 21:17:19

Last modified on

2013-05-16 21:17:19

Owner

marijke (8873)

Last modified by

unlord (1)

Numerical id

7

Author

marijke (1)

Entry type

Definition

Classification

msc 05C15

Related topic

ColoringsOfPlaneGraphs

Related topic

ProofOfVizingsTheoremForGraphs

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