MATH 682
Notes
Combinatorics and Graph Theory II
1
Other coloring problems
1.1
Edge coloring
Many properties which are traditionally based on vertices (e.g. connectivity) also exist in an edge
version, so it is probably not surprising that one can investigate coloring as a property of edges as
well as vertices.
Deﬁnition 1.
A
r
edgecoloring
of a graph
G
is a function
c
:
E
(
G
)
→ {
1
,
2
,
3
,...,r
}
such that,
if
e
and
f
are distinct edges sharing an endpoint,
c
(
e
)
6
=
c
(
f
). If
G
has an
r
edgecoloring, it is
called
r
edgecolorable
. The
edge chromatic number
of
G
, denoted
χ
0
(
G
), is the least
r
such that
G
is
r
edgecolorable.
It is easy to ﬁnd a lower bound for
χ
0
(
G
), and a quite sloppy upper bound:
Proposition 1.
For a nontrivial graph
G
,
Δ(
G
)
≤
χ
0
(
G
)
≤
2Δ(
G
)

1
.
Proof.
Let
v
be a vertex of maximum degree in
G
, i.e.
d
(
v
) = Δ(
G
), so
v
has incident edges
e
1
,e
2
,...,e
Δ(
G
)
. All of these edges must be diﬀerent colors in a valid coloring, so at least Δ(
G
)
colors are required.
On the other hand, if we enact a greedy coloring of the edges (i.e. coloring the edges, in some order,
using the least color possible at each step), then each edge
{
u,v
}
which we have not yet colored is
forbidden to use at most (
d
(
u
)

1) + (
d
(
v
)

1)
≤
2Δ(
G
)

2 colors. Thus, there is some color in
{
1
,
2
,...,
2Δ(
G
)

1 which is not forbidden, and can be used to color this edge in a greedy coloring.
Since this is true of every edge we consider for greedy coloring, the entire greedy coloring can be
achieved with 2Δ(
G
)

1 or fewer colors.
There is a whole family of graphs that achieves the lower bound seen above:
Theorem 1
(K¨onig ’18)
.
If
G
is bipartite, then
χ
0
(
G
) = Δ(
G
)
.
Proof.
We shall prove this via induction on
k
G
k
. When
k
G
k
= 0, both
χ
0
(
G
) and Δ(
G
) are zero,
so the base case is trivially true.
Now, considering an arbitrary
G
, let us select an edge
e
=
{
u,v
}
. By our inductive hypothesis,
G

e
is Δ(
G
)colorable (in fact it is possible that one color is not even necessary, but we don’t
need that fact). Since
d
G

e
(
u
)
≤
Δ(
G
)

1 and
d
G

e
(
v
)
≤
Δ(
G
)

1, then there must be some
color
a
not utilized on edges incident on
u
, and some color
b
not utilized on edges incident to
v
. If
a
=
b
, we may extend our coloring of
G

e
, coloring
e
in the color
a
to get a simple Δ(
G
)coloring
of
G
. The interesting and troublesome case, then, is where the color not utilized at
u
is utilized at
v
and vice versa.
Let us denote
v
=
v
0
, and denote the other endpoint of the edge in color
a
from
v
0
as
v
1
. Now
note that if
v
1
has no incident edge in color
b
, then we could simply change the edge
{
v
0
,v
1
}
to be
of color
b
, and then we would have the situation described above, where the coloring in
G

e
is
easily extended to
G
. Now, consider the possibility that
v
1
does have an incident edge in color
b
,
and denote its other endpoint as
v
2
. Then, if
v
2
had no incident edge in color