Algorithmic Graph Theory
Sommer Term 2015
Lecture #9 (part II)
Coloring Planar Graphs
Prof. Rajasekhar Inkulu
IIT Guwahati
Coloring (Planar) Graphs
Def.
Let G = (V , E ) be a graph.
A map f : V → {1, . . . , k } is a k -coloring
if, for each edge uv ∈ E , it holds that f (u ) 6= f (v ).
Obs.
G bipartite ⇔ G 2-colorable.
G k -partite ⇔ G k -colorable.
Obs.
Every planar graph is 6-colorable.
Proof.
G has a vertex v of degree ≤ 5.
Color G − v inductively. Take 6th color for v .
v
G −v
Thm. Four-Color Theorem
Every planar graph is 4-colorable.. . .
[Appel & Haken 1976]
[Robertson, Sanders, Seymour, Thomas 1997]
. . . and there are planar graphs that need 4 colors.
The Five-Color Theorem of 1890
Thm. Five-Color Theorem
[Heawood 1890]
Every planar graph is 5-colorable.
Def.
Given a graph G = (V , E ) and, for each
vertex v of G , a list Lv of “colors”,
S
a list coloring of G is a map λ : V → v Lv
s.t. • λ(v ) ∈ Lv and
• λ(u ) 6= λ(v ) for each uv ∈ E .
Percy John Heawood
1861 Newport, GB
1955 Durham, GB
Obs. A “normal” coloring c : V → {1, . . . , k } is a list coloring
with Lv = {1, . . . , k } for every v ∈ V .
Expl.
1,2,3
2,3,4
3,4
1,3,4
1,2,4
1,2
Does this graph have a coloring
with the given lists?
Yes!
List Colorability
Def.
A graph G = (V , E ) is k -list colorable
if G has a list coloring for any choice of lists of length k .
⇐
Obs. G k -list colorable ⇒ k -colorable.
Expl. Every bipartite graph is 2-colorable –
but not necessarily 2-list colorable.
List Colorability of Planar Graphs
Thm. Not-Four-Color Theorem
Not every planar graph is 4-list colorable.
[Voigt 1993]
Margit Voigt
HTW Dresden
Carsten Thomassen
1948 in Grindsted, DK
Thm. Every planar graph is 5-list colorable. [Thomassen 1994]
(and hence also 5-colorable!)
Proof of Thomassen’s Theorem
Wlog., G is nearly triangulated, i.e. all interior faces are triangles
and the boundary of the
x
outer face is elementary.
α
Thomassen’s trick:
β
y
Strengthen claim,
prove by induction!
Claim. Let G nearly triang., K boundary of the outer face, and
(i) two adj. vertices x , y ∈ K are colored with α 6= β.
(ii) |Lv | ≥ 3 for every v ∈ K \ {x , y }
(iii) |Lv | ≥ 5 for every v ∈ V \ K .
then the coloring of x and y can be extended to G .
Proof. By induction on n = |V |.
z
n = 3:
Color z with γ ∈ Lz \ {α, β}.
x
y
X
K1
x G1
Induction Step (n > 3)
K2
G2
y
Case 1: K has a chord uv .
v
u
uv splits K into K1 and K2 .
Let G1 be the subgraph of G inside K1 + uv (incl. boundary).
Apply induction hypothesis (IH) to G1 .
X
w
Apply IH (with u and v already being colored!) to G2 .
Case 2: K has no chord.
Let w 6= y be a neighbor of x on K .
Let N (w ) = {x , w1 , . . . , wt , v } be the neighborhood of w .
|Lw | ≥ 3 ⇒ there are γ, δ ∈ Lw \ {α}
x
α
β
v
...
wt
w
1
y
G0
L0wi := Lwi \ {γ, δ}. G 0 = G − w is nearly triangulated.
⇒ G 0 with lists L0 fulfills IH
⇒ 5-list coloring of G 0
Color w with {γ, δ} \ color(v ) ⇒ 5-list coloring of G
X