We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this work we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm that constructs a 4-clique coloring of it.Facultad de Ciencias Exacta

Bonomo, Flavia

Mazzoleni, María Pía

Stein, Maya

English

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B1-EPG GRAPHS ARE 4-CLIQUE COLORABLE
Flavia Bonomo†, Marı´a Pı´a Mazzoleni‡ and Maya Stein†‡
†CONICET and Departamento de Computacio´n, FCEN-UBA, Buenos Aires, Argentina, fbonomo@dc.uba.ar
‡CONICET and Departamento de Matema´tica, FCE-UNLP, La Plata, Argentina, pia@mate.unlp.edu.ar
†‡Departamento de Ingenierı´a Matema´tica, Universidad de Chile, Santiago, Chile, mstein@dim.uchile.cl
Abstract: We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no
(maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this
work we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one
bend) are 4-clique colorable. Moreover, given a B1-EPG representation of a graph, we provide a linear time algorithm
that constructs a 4-clique coloring of it.
Keywords: clique coloring, edge intersection graphs, paths on grids, polynomial time algorithm.
2000 AMS Subject Classification: 05C15 - 05C62
1 INTRODUCTION
An EPG representation of a graph G, hP,Gi, is a collection of paths P of the two-dimensional grid G
where the paths represent the vertices ofG in such a way that two vertices ofG are adjacent inG if and only
if the corresponding paths share at least one edge of the grid. A graph which has an EPG representation is
called an EPG graph (EPG stands for Edge-intersection of Paths on a Grid). In this work, we consider the
subclass B1-EPG.
A B1-EPG representation of G is an EPG representation in which each path in the representation has at
most one bend (turn on a grid point). Recognizing B1-EPG graphs is an NP-complete problem (see [9]).
EPG graphs have a practical use, for example, in the context of circuit layout setting, which may be
modelled as paths (wires) on a grid. In the knock-knee layout model, two wires may either cross or bend
(turn) at a common point grid, but are not allowed to share a grid edge; that is, overlap of wires is not
allowed. In this context, some of the classical optimization graph problems are relevant, for example,
maximum independent set and coloring. More precisely, the layout of a circuit may have multiple layers,
each of which contains no overlapping paths. Referring to a corresponding EPG graph, then each layer is
an independent set and a valid partitioning into layers corresponds to a proper coloring.
In this work, we consider the problem of clique coloring, that is, coloring the vertices of a given graph
such that no (maximal) clique of size at least two is monocolored. We prove thatB1-EPG graphs are 4-clique
colorable. Moreover, given a B1-EPG representation of a graph, we provide a polynomial time algorithm
that constructs a 4-clique coloring of it.
2 PRELIMINARIES
In this work all graphs are connected, finite and simple. Notation we use is that used by Bondy and
Murty [2]. The vertex set of a graph G is the set of all the vertices of G, denoted by V (G), and the edge set
of G is the set of all the edges of G, denoted by E(G).
A complete is a set of pairwise adjacent vertices. A clique is a complete which is not properly contained
in another complete.
A k-coloring of a graph G is a function f : V (G) → {1, 2, . . . , k} such that adjacent vertices have
different labels, i.e., no complete of size 2 is monocolored. The chromatic number of a graph G is the
smallest positive integer k such that G has a k-coloring, denoted by χ(G). A k-clique coloring of a graph
G is a function f : V (G)→ {1, 2, . . . , k} such hat no clique of G with size at least two is monocolored. A
graphG is k-clique colorable ifG has a k-clique coloring. The clique chromatic number ofG is the smallest
k such that G has a k-clique coloring, denoted by χc(G).
Clique coloring has some similarities with usual coloring, for example, any coloring is also a clique
coloring, and optimal colorings and clique colorings coincide in the case of triangle-free graphs. But there
are also essential differences, for example, a clique coloring of a graph may not be a clique coloring for
VI MACI 2017 - Sexto Congreso de Matemática Aplicada, Computacional e Industrial
2 al 5 de mayo de 2017 – Comodoro Rivadavia, Chubut, Argentina
167
its subgraphs. Subgraphs may have a greater clique chromatic number than the original graph. Another
difference is that even a 2-clique colorable graph can contain an arbitrarily large clique. It is known that the
2-clique coloring problem is NP-complete, even under different constraints [1, 10].
Many families of graphs are 3-clique colorable, for example, comparability graphs, co-comparability
graphs, circular arc graphs and the k-powers of cycles [3, 4, 6, 7]. In [1], Bacso et al. proved that almost
all perfect graphs are 3-clique colorable and conjectured that all perfect graphs are 3-clique colorable. On
the other hand, some families of graphs have unbounded clique chromatic numbers, for example, triangle-
free graphs, UE graphs and line graphs [1, 5, 11]. It has been proved that chordal graphs, and in particular
interval graphs, are 2-clique colorable [12].
Moreover, every chordal graph admits a 2-clique coloring in which one of the color classes is an inde-
pendent set. Such a coloring is obtained by the following algorithm: let v1, . . . , vn be a perfect elimination
ordering of the vertices of a chordal graph G, i.e., for each i, N [vi] is a clique of G[{vi, . . . , vn}]; color the
vertices from vn to v1 with colors a and b in such a way that vn gets color a and vi gets color b if and only
if all of its neighbors that are already colored got color a.
3 B1-EPG GRAPHS ARE 4-CLIQUE COLORABLE
In this Section, we prove that B1-EPG graphs are 4-clique colorable. In order to prove the main result of
this work we need the following definitions and theorem.
The claw is a star with three edges (see Figure 1). The edges of the star are called the legs.
Figure 1: The claw.
Let hP,Gi be a B1-EPG representation of a graph G. For any edge e in the grid G, let P[e] = {P ∈
P/e ∈ P}. For any clawH in G, let P[H]= {P ∈ P/ P contains two legs of the clawH}. If the collection
P[e] corresponds to a clique C in G, then C is called an edge clique. Similarly, if P[H] corresponds to a
clique C in G, then C is called a claw clique (it holds if every pair of legs of H is contained in a path P of
P). We say that a claw clique is centered at x if the corresponding claw is a star with center x.
Figure 2: An EPG representation of the 3-sun. The central triangle {2, 3, 5} is a claw clique; the other three triangles
are edge cliques.
Theorem 1 [8] Let hP,Gi be a B1-EPG representation of a graph G. Every clique in G corresponds to
either an edge clique or a claw clique in hP,Gi.
The proof of the following theorem have been omitted for lack of space.
Theorem 2 Let G be a B1-EPG graph. Then, G is 4-clique colorable. Moreover, a 4-clique coloring of G
can be obtained in linear time on the number of vertices and edges, given a B1-EPG representation of G.
VI MACI 2017 - Sexto Congreso de Matemática Aplicada, Computacional e Industrial
2 al 5 de mayo de 2017 – Comodoro Rivadavia, Chubut, Argentina
168
4 CONCLUSION
In this work we have proved that B1-EPG graphs are 4-clique colorable, and that such coloring can be
obtained in polynomial time, given a B1-EPG representation. We conjecture that indeed B1-EPG graphs
are 3-clique colorable. An example of a B1-EPG graph that requires 3 colors for a clique coloring is the
chordless cycle on 5 vertices.
REFERENCES
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[9] D. HELDT, K. KNAUER AND T. UECKERDT, Edge-intersection graphs of grid paths: the bend number, Discrete Applied
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