Vizing’s theorem

Vizing’s theorem

\PMlinkescapephrase

class number

Definitions and notation

In a graph or

multigraph

G

, let

ρ

⁢

(

v

)

denote the

valency

of vertex
(node)

v

, and let

ρ

⁢

(

G

)

denote the largest valency in

G

(often written
just

ρ

on its own,

Δ

⁢

(

G

)

is another common notation).

Let the

multiplicity

μ

⁢

(

v

,

w

)

of vertices (nodes)

v

and

w

be the number of parallel edges that link them, and here too let

μ

⁢

(

G

)

be
the largest multiplicity in

G

. A graph is a multigraph for which

μ

⁢

(

G

)

=

1

.

An

edge coloring

of a (multi)graph

G

is a

mapping

from the
set

E

of its edges to a set

K

of items called colors, in such a way
that at any vertex

v

, the

ρ

⁢

(

v

)

edges there all have a different
color. An

edge-

k

-

coloring

is an edge-coloring where

|

K

|

=

k

.

Note that a loop (an edge joining

v

to itself) accounts for two of the

ρ

⁢

(

v

)

edges at

v

and cannot have a color different from itself,
so edge-colorings as defined here do not exist for

pseudographs

(

structures

that are allowed to have loops).

The

chromatic index

(aka

edge-chromatic number

)

χ

′

⁢

(

G

)

is
the smallest number

k

for which an edge-

k

-coloring of

G

exists. This now
standard notation is analogous to the

chromatic number

χ

⁢

(

G

)

which
refers to

vertex coloring

.

Vizing’s theorem

For any multigraph

G

(this includes graphs), we have of course

χ

′

⁢

(

G

)

⩾

ρ

⁢

(

G

)

immediately from the definition. Surprisingly though, we also have

Theorem

(Vizing)

For any graph

G

,

χ

′

⁢

(

G

)

⩽

ρ

⁢

(

G

)

+

1

This celebrated theorem was proved by V. G. Vizing in 1964 while still a
graduate student (at Novosibirsk). It is a special case of

Theorem (Vizing)

For any multigraph

G

,

χ

′

⁢

(

G

)

⩽

ρ

⁢

(

G

)

+

μ

⁢

(

G

)

(edge-coloring of multigraphs is the subject of Vizing’s doctoral thesis, 1965).
The theorem was proved independently by R. P. Gupta in 1966; nowadays various
versions of the proofs exist. A key rôle is played by

connected

subgraphs

H

⁢

(

a

,

b

)

of colors

a

and

b

that cannot be extended further. They are analogous to

Kempe chains

in face or vertex colorings, but have a simpler structure: they are either paths or

closed paths

(the latter of even length).

http://planetmath.org/node/6932

See here for a proof (in the case of graphs).

Ore

[

Ore67

]

gives a sharper bound for multigraphs. Let the

enlarged valency

ρ

+

⁢

(

v

)

be given by

ρ

+

⁢

(

v

)

=

def

ρ

⁢

(

v

)

+

max

w

⁡

μ

⁢

(

v

,

w

)

where

w

can be taken over all vertices of the graph; it effectively only
ranges over those

adjacent

to

v

(as

μ

⁢

(

v

,

w

)

is zero for the others).
Again, let

ρ

+

⁢

(

G

)

denote the maximum enlarged valency occurring in

G

. Now

Theorem (Ore)

For any multigraph

G

,

χ

′

⁢

(

G

)

⩽

ρ

+

⁢

(

G

)

Shannon’s theorem

The following theorem was proven in 1949 by
Claude E. Shannon. He also gives examples, for any value of

ρ

, of
multigraphs that actually attain the bound,
the so-called Shannon graphs: they have three vertices, with

μ

⁢

(

a

,

b

)

=

μ

⁢

(

a

,

c

)

=

⌊

ρ

/

2

⌋

and

μ

⁢

(

b

,

c

)

=

⌈

ρ

/

2

⌉

.

Theorem (Shannon)

For any multigraph

G

,

χ

′

⁢

(

G

)

⩽

⌊

3

2

⁢

ρ

⁢

(

G

)

⌋

While giving a much worse bound for graphs, it gives for some multigraphs a
better bound than Vizing’s theorem. Nevertheless, it is also possible to prove
it from the latter

[

FW77

]

.

Only in the

context

of

graph colorings

is

Shannon’s theorem

understood
to refer to the one here; in the wider world the term tends to refer to any of
his fundamental theorems in information theory.

Here too Ore

[

Ore67

]

gives a sharper bound based on the maximum of
a local expression. Let

σ

⁢

(

v

)

be 0 if

v

has fewer than two
neighbours, and otherwise

σ

⁢

(

v

)

=

def

max

v

′

,

v

′′

⁡

(

ρ

⁢

(

v

)

+

ρ

⁢

(

v

′

)

+

ρ

⁢

(

v

′′

)

)

where

v

′

and

v

′′

range over all pairs of distinct neighbours of

v

. Again, let

σ

⁢

(

G

)

be its
largest value in the graph.

Theorem (Ore)

For any multigraph

G

,

χ

′

⁢

(

G

)

⩽

max

⁡

(

ρ

⁢

(

G

)

,

⌊

1

2

⁢

σ

⁢

(

G

)

⌋

)

Chromatic class

Vizing’s theorem has the effect of placing each multigraph

G

in one of the
classes 0, 1, 2, …

μ

⁢

(

G

)

where the class number is

χ

′

⁢

(

G

)

-

ρ

⁢

(

G

)

.

For graphs it means they split into just two classes: those that can be
edge-colored in

ρ

⁢

(

G

)

colors and those that need

ρ

⁢

(

G

)

+

1

colors. The logical name for them would be class 0 and class 1; unfortunately
the standard terminology is class 1 and 2 (or I and II).

Class I

(graphs that can be edge-

ρ

-colored) contains among others

•

single

n

-cycles for even

n

†

•

complete graphs

K

n

for even

n

‡

•

bipartite graphs

(this is König’s theorem, 1916)

•

bridgeless planar trivalent graphs (

four-color theorem

via

Tait coloring

)

•

planar graphs

with

ρ

⁢

(

G

)

⩾

8

(by another theorem of Vizing)

•

planar graphs with

ρ

⁢

(

G

)

⩾

6

?? (conjecture of Vizing)

Class II

(graphs that need

ρ

+

1

colors) contains among others

•

single

n

-cycles for odd

n

†

•

complete graphs

K

n

for odd

n

‡

•

K

n

for odd

n

with a few edges missing (by a couple of theorems)

•

trivalent graphs with a bridge

¶

but the general

classification

problem has thus far eluded the best efforts of
Vizing and many others. There are interesting links here with polyhedral
decompositions (aka cyclic double covers) and

embeddings

in

surfaces

.

A(n

edge-

)

critical graph

is a

connected graph

of class II but such
that removing any of its edges makes it class I. As often in

graph theory

,
such a minimality condition imposes a certain amount of structure on the graph.
There are conjectures…

Almost all graphs are in class I

Let

#

⁢

G

n

be the number of graphs with

n

vertices,
and

#

⁢

G

n

I

the number of them in class I.

Theorem (P. Erdős and R. J. Wilson)

lim

n

→

∞

⁡

#

⁢

G

n

I

#

⁢

G

n

=

1

So graphs of class II get

relatively

rarer for larger graph sizes. The

absolute

numbers do still increase. For

cubic graphs

for

instance

, we saw every one with a bridge is in class II. Bridgeless cubic graphs of class II are a bit thinner on the ground. By the four-color theorem, via Tait coloring, we know all of them are non-planar. Rarer still are those of them with girth at least five (and some non-triviality conditions); they are so hard to find that Martin Gardner dubbed them

snarks

. The

Petersen graph

is one, and a few

infinite

families of snarks have been found.

†

ρ

=

2

, so an edge-

ρ

-coloring must use

alternating

colors. For odd

n

that’s impossible.

‡

ρ

=

n

-

1

, note it is also the valency of every vertex.
Fix two colors

a

and

b

. A Kempe chain

H

⁢

(

a

,

b

)

can only terminate at a vertex where one of those colors is missing but in

K

n

we cannot afford to miss any color at any vertex, so every

H

⁢

(

a

,

b

)

is a cycle of even length and together they visit all

n

vertices. For odd

n

that’s impossible.
For even

n

there are ways to

construct

the coloring (try it).

¶

ρ

=

3

and again the valency of every vertex. For the same
reason as in the previous note, every

H

⁢

(

a

,

b

)

or

H

⁢

(

c

,

a

)

or

H

⁢

(

b

,

c

)

is a cycle. Every edge must be in two such, but a bridge cannot be part of a cycle.

References

1

Ore67

Oystein Ore

,

The

Four-Color Problem

,

Acad. Pr. 1967,

ISBN  0 12 528150 1

Long the standard work on its subject, but written before
the theorem was proven. Has a wealth of other graph theory
material, including proofs of (improvements of) Vizing’s and
Shannon’s theorems.

FW77

S. Fiorini

and

R. J. Wilson

,

Edge-colourings of graphs

, Pitman 1977,

ISBN  0 273 01129 4

The first ever book devoted to edge-colorings,
including material previously found only in Russian

language

journal articles. Has proofs of Vizing’s and
Shannon’s theorems.

SK77

Thomas L. Saaty

and

Paul C. Kainen

,

The Four-Color Problem: assaults and conquest

,

McGraw-Hill 1977; repr. Dover 1986,

ISBN  0 486 65092 8

Wonderfully broad, not only focussing on the usual
route to the Appel-Haken proof but also giving lots of
other material.

Wil02

Robert A. Wilson

,

Graphs, Colourings and the Four-colour Theorem

,
Oxford Univ. Pr. 2002,

ISBN  0 19 851062 4

(pbk),

http://www.maths.qmul.ac.uk/ raw/graph.htmlhttp://www.maths.qmul.ac.uk/ raw/graph.html

(errata &c.)

A good general course in graph theory, with special focus on
the four-color theorem and details of the Appel-Haken proof.
Has proof of Vizing’s theorem.

Title

Vizing’s theorem

Canonical name

VizingsTheorem

Date of creation

2013-03-22 15:10:36

Last modified on

2013-03-22 15:10:36

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

11

Author

marijke (8873)

Entry type

Theorem

Classification

msc 05C15

Related topic

TaitColoring

Defines

edge coloring

Defines

edge-

k

-coloring

Defines

chromatic index

Defines

edge-chromatic number

Defines

class I

Defines

class II

Defines

class 1

Defines

class 2

Defines

edge-critical graph

Defines

enlarged valency

Defines

snark

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