Choosability of P
5
-free graphs
∗
Petr A. Golovach
†
Pinar Heggernes
†
Abstract
A graph is k-choosable if it admits a proper coloring of its vertices
for every assignment of k (possibly different) allowed colors to choose
from for each vertex. It is NP-hard to decide whether a given graph is
k-choosable for k ≥ 3, and this pr oblem is co nsidered strictly harder
than the k-coloring problem. Only few positive results are known on
input graphs with a given structure . Here, we prove that the problem
is fixed para meter tractable on P
5
-free graphs when parameterized by
k. This graph class contains the well known and widely studied class of
cographs. Our result is s urprising since the pa rameterized complexity
of k-coloring is still open on P
5
-free graphs. To give a complete picture,
we show that the pr oblem remains NP-hard on P
5
-free graphs when k
is a part of the input.
1 Introduction
Graph coloring is one of the most well known and intensively studied prob-
lems in graph theory. The k-Coloring problem asks whether the vertices
of an input graph G can be colored with k colors such that no pair of ad-
jacent vertices receive the same color (such coloring is also called a proper
coloring). This problem is known to be NP-complete even when k ≥ 3 is
not a part of the input but a fixed constant.
Vizing [19] and Erd˝os et al. [6] introduced a version of graph coloring
called list coloring. In list coloring, a set L(v) of allowed colors is given for
each vertex v of the input graph, and we want to decide whether a proper col-
oring of the graph exists such that each vertex v receives a color from L(v).
If G has a list coloring for every assignment of lists of cardinality k to its ver-
tices, then G is said to be k-choosable. Hence the k-Choosability problem
∗
This work is supported by the Research Council of Norway.
†
Department of Informatics, University of Bergen, N-5020 Bergen, Norway. Emails:
{Peter.Golovach|Pinar.Heggernes}@ii.uib.no
1
asks whether an input graph G is k-choosable. List coloring has received
increasing attention since the beginning of 90’s, and there are very good sur-
veys [1, 17] and books [11] on the subject. It is proved to be a very difficult
problem; Gutner and Tarsi [9] proved that k-Choosability is Π
P
2
-complete
for bipartite graphs for any fixed k ≥ 3, w hereas 2-Choosability can be
solved in polynomial time [6]. The 3-Choosability and 4-Choosability
problems remain Π
P
2
-complete for planar graphs, whereas any planar graph
is 5-choosable [16]. Due to these hardness results, upto the assumption that
NP is not equal to co-NP, Choosa bility is strictly harder than Coloring
on general graphs [1].
Despite being a difficult problem to deal with, Choosability has app li-
cations in a large variety of areas, like various kinds of scheduling problems,
VLSI design, and frequen cy assignments [1]. Consequently, any attempt to
solve this problem is of interest, and we attack it using structural inf ormation
on the input and parameterized algorithms. A problem is fixed parameter
tractable (FPT) if its input can be partitioned into a main part (typically
the input graph) of size n and a p arameter (typically an integer) k so that
there is an algorithm that solves the problem in time O(n
c
· f(k)), w here
f is a computable function dependent only on k, and c is a fixed constant
independent of input [5]. In this case, we say that the problem is FPT w hen
parameterized by k. The field of parameterized algorithms and fixed pa-
rameter complexity/tractability has been flourishing during the last decade,
with many new results appearing every year in high level conferences and
journals, and it has been enriched by several new books [7, 14].
In this paper, we show that k-Choosability is fixed parameter tractable
on P
5
-free graphs. These are graphs containing no induced copy of a simple
path on 5 vertices, and this graph class contains the class of cographs that
has been subject to extensive theoretical study [3]. An interesting point
to mention is that the fixed parameter tractability of k-Coloring on P
5
-
free graphs is still open [10]. As mentioned above, Choosability is more
difficult than Coloring on general graphs. Our result indicates that the
opposite might be true for the class of P
5
-free graphs. In last year’s MFCS,
Ho`ang et al. showed that k-Coloring can be solved in polynomial time for
any fixed k on P
5
-free graph s [10], but in their running time k contributes
to the degree of th e polynomial. Furtherm ore, k-Coloring is NP-complete
on P
5
-free graphs when k is a part of input [12]. To give a complete picture,
here we show that k-Choosability is NP-hard on P
5
-free graphs when k is
a part of input. Thus fixed parameter tractability is the best we can expect
to achieve for k-Choosability on this graph class.
To mention other existing results on th e coloring problem on graphs
2
that do not contain long induced paths, 3-Coloring has a polynomial-
time solution on P
6
-free graphs [15], 5-Coloring is NP-complete for P
8
-free
graphs, and 4-Coloring is NP-complete for P
12
-free graphs [20].
2 Definitions and preliminaries
We consider finite undirected graphs without loop s or multiple edges. A
graph is denoted by G = (V, E), wh ere V = V (G) is the set of vertices and
E = E(G) is the set of edges. For a vertex v ∈ V , the set of vertices that
are adjacent to v is called the neighborhood of v an d den oted by N
G
(v) (we
may omit index if the graph under consideration is clear from the context).
The degree of a vertex v is deg(v) = |N(v)|. The average degree of G is
d(G) =
1
|V |
P
v∈V
deg(v). For a vertex subset U ⊆ V the subgraph of G
induced by U is denoted by G[U]. A set U ⊆ V is a clique if all vertices
in U are pairwise adjacent in G. A set of vertices U is a dominating set if
for each vertex v ∈ V , either v ∈ U or there is a vertex u ∈ U such that
v ∈ N(u). We also say that a subgraph H of G is dominating if V (H) is a
dominating set. We denote by G − U the graph G[V \ U ], and by G − u the
graph G[V \ {u}] for u ∈ V .
A vertex coloring of a graph G = (V, E) is an assignment c: V → N of
a positive integer (color) to each vertex of G. The coloring c is proper if
adjacent vertices receive distinct colors. Assume that each vertex v ∈ V is
assigned a color list L(v) ⊂ N, which is the set of admissible colors for v. A
mapping c: V → N is a list coloring of G if c is a proper vertex coloring and
c(v) ∈ L(v) for every v ∈ V . For a positive integer k, G is k-choosable if G
has a list coloring for every assignment of color lists L(v) with |L(v)| = k
for all v ∈ V . The choice number (also called list chromatic number) of G,
denoted ch(G), is the minimum integer k such that G is k-choosable. The
k-Choosability problem asks for a given graph G and a positive integer k,
whether G is k-choosable. It is known that dense graphs have large choice
number [1], as indicated by the following result.
Proposition 1 ([1]). Let G be a graph and s be an integer. If
d(G) > 4
s
4
s
log(2
s
4
s
)
then ch(G) > s.
By P
n
we denote the graph on vertex set {v
1
, v
2
, . . . , v
n
} and edge set
{v
1
v
2
, v
2
v
3
, . . . , v
n−1
v
n
}. A graph is P
n
-free if it does not contain P
n
as
3
an induced subgraph. Cographs are the class of P
4
-free graphs, and they
are contained in th e class of P
5
-free graphs. These graph classes can be
recognized in polynomial time. The f ollowing stru ctural property of P
5
-free
graphs was proved by Bacs´o and Tuza [2].
Proposition 2 ([2]). Every connected P
5
-free graph has either a dominating
clique or a dominating P
3
.
It follows from the results of [2] that such a clique or path can be con-
structed in polynomial time.
Finally, we distinguish between the parameterized and the non-
parameterized versions of our problem. In the Choosability problem, G
and k are input. We den ote by k-Choosab ility the version of the problem
parameterized by k.
3 k-Choosability is FPT on P
5
-free graphs
In this section we prove that k-Choosab ility is fixed parameter tractable
on P
5
-free graphs.
Theorem 1. The k-Choosability problem is FPT on P
5
-free graphs.
Proof. We give a constructive proof of this theorem by describing a recurs ive
algorithm based on Propositions 1 and 2 that checks whether ch(G) ≤ k.
We assume th at k ≥ 3, since for k ≤ 2, k-Choosability can be solved in
polynomial time for general graphs [6]. If G is disconnected, then ch(G) is
equal to the maximum choice number of the connected components of G.
Thus we also assume that G is connected.
Our algorithm uses as its main tool a procedure called Color, given
in Algorithm 1. This procedure takes as input a connected P
5
-free graph
G and a set W = {w
1
, . . . , w
r
} ⊆ V (G) with a sequence of color lists L =
(L(w
1
), . . . , L(w
r
)), each of size k. For the notation in this procedure, we let
L = L(w
1
)∪· · ·∪L(w
r
), and we denote l = max{max L(w
1
), . . . , max L(w
r
)}.
Let also L = L(w
1
)×· · ·×L(w
r
) and X = 2
L
. We say that vertices w
1
, . . . , w
r
are colored by c = (c
1
, . . . , c
r
) ∈ L if each w
i
is colored by c
i
. Set H = G−W .
Procedure Color produces an output which either contains a list of different
sets X = (X
1
, . . . , X
s
), X
i
∈ X, such that for any assignment of color lists
of size k to vertices of H, there is a set X
i
with the property that any
c ∈ X
i
can be used for coloring of W with respect to adjacencies between
vertices in W and vertices in V (H), or the outp ut contains ”NO” if there is
a list assignment for vertices of H such that no list coloring exists. Denote
4
Procedure Color(G, W , L)
Find a dominating set U = {u
1
, . . . , u
p
} of H = G − W , such that U is
a clique or U induces a P
3
;
Let X = ∅;
if p > k then Return(NO), Halt;
if d(G[W ∪ U]) > d then Return(NO), Halt;
forall Color lists L(u
1
), . . . , L(u
p
) ⊆ {1 , . . . , l, l + 1, . . . , l + kp}, s.t.
|L(u
i
)| = k do
if U = V (H) then
Let X = ∅;
forall List colorings s of H do
Let X := X ∪ {c ∈ L : c(w
i
) 6= s(u
j
) if w
i
u
j
∈ E(G)};
if X 6= ∅ then Add(X , X); else Return(NO), Halt;
if U 6= V (H) then
Let H
1
, . . . , H
q
be the connected components of H − U, and
let F
i
= G[W ∪ U ∪ V (H
i
)] for i ∈ {1, . . . , q};
Let L
′
= (L(u
1
), . . . , L(u
p
)), L
′
= L × L(u
1
) × · · · × L(u
p
);
for i = 1 to q do
Color(F
i
, W ∪ U, L ∪ L
′
);
if the output is NO then
Return(NO), Halt;
else
Let X
i
be the output;
Let Y = X
1
;
for i = 2 to q do
Let Z = ∅;
forall X ∈ X
i
and Y ∈ Y do
if X ∩ Y 6= ∅ then Add(Z, X ∩ X
′
);
else Return(NO), Halt;
Let Y = Z;
forall Z ∈ Z do
Let X = {(c(w
1
), . . . , c(w
r
)): c ∈ Z, c(w
i
) 6= c(u
j
) if w
i
u
j
∈
E(G) and c(u
i
) 6= c(u
j
) if u
i
u
j
∈ E(G)};
if X 6= ∅ then Add(X , X);
else Return(NO), Halt;
if X = ∅ then Return(NO), Halt; else Return(X ).
Algorithm 1: Pseudo code for th e procedure Color
5