Circular chromatic number of triangle-free hexagonal graphs Circular chromatic number of triangle-free hexagonal graphs

Circular chromatic number of triangle-free

hexagonal graphs

1 Petra ˇ Sparl and 2 Janez ˇ Zerovnik

1 University of Maribor

Faculty of Civil Engineering

Smetanova 17

SI-2000 Maribor, SLOVENIA

2 University of Maribor

Faculty of Mechanical Engineering

Smetanova 17

SI-2000 Maribor, SLOVENIA

1,2 IMFM

SI-1000 Ljubljana, SLOVENIA.

Abstract

An interesting connection between graph homomorphisms to odd cycles

and circular chromatic number is presented. By using this connection,

bounds for circular chromatic number of triangle-free hexagonal graphs

(i.e. induced subgraphs of triangular lattice) are given.

Keywords: graph homomorphism, circular chromatic number, triangle-free

hexagonal graph

2000 Mathematics Subject Classification: 05C15, 68R10

1 Introduction

Suppose G and H are graphs. A homomorphism from G to H is a mapping f

from V (G) to V (H) such that f(x)f(y) ∈ E(H) whenever xy ∈ E(G). Homomorphisms

of graphs are studied as a generalization of graph colorings. Indeed,

a vertex coloring of a graph G with n-colors is equivalent to a homomorphism

from G to Kn. Therefore, the term H-coloring of G has been employed to describe

the existence of a homomorphism of a graph G into the graph H. In such

a case graph G is said to be H-colorable. Graph homomorphism were widely

studied in different areas, see [2, 3] and the references there. One of the approaches

is deciding whether an arbitrary graph G has a homomorphism into a

1

fixed graph H. The main result, regarding the complexity of H-coloring problem,

was given by Hell and Neˇsetril in 1990 . They proved that H-coloring

problem is NP-complete, if H is non-bipartite graph and polynomial otherwise.

Several restrictions of the H-coloring problem have been studied . One of the

restricted H-coloring problems was studied in , where H is an odd cycle and

G an arbitrary, the so-called, hexagonal graph, which is an induced subgraph of

a triangular lattice. It was shown that any triangle-free hexagonal graph G is

C5-colorable. This result will be used in section 4 to obtain upper bounds for

circular chromatic number of triangle-free hexagonal graphs.

Another interesting approach regarding homomorphisms can be found in the

literature. In  author discuses the connection between graph homomorphisms

and so called circular colorings. A partial result of this connection in a slightly

different form is given in Section 3.

Circular coloring and circular chromatic number are natural generalizations

of ordinary graph coloring and chromatic number of a graph. The circular

chromatic number was introduced by Vince in 1988, as ”the star-chromatic

number” . Here we present an equivalent definition of Zhu .

Definition 1 Let C be a circle of (Euclidean) length r. An r-circular coloring

of a graph G is a mapping c which assigns to each vertex x of G an open unit

length arc c(x) of C, such that for every edge xy ∈ E(G), c(x) ∩ c(y) = ∅. We

say a graph G is r-circular colorable if there is an r-circular coloring of G. The

circular chromatic number of a graph G, denoted by χc(G), is defined as

χc(G) = inf{r : G is r-circular colorable}.

For finite graphs G it was proved [1, 6, 7] that the infimum in the definition of

the circular chromatic number is attained, and the circular chromatic numbers

χc(G) are always rational.

In this paper we present a connection between homomorphisms to odd cycles

and circular chromatic number. Using this connection we prove:

• For an arbitrary graph G the following two statements are equivalent:

(i) k is the biggest positive integer for which there exists a homomorphism

f : G → C2k+1,

(ii) 2 + 1

k+1 < χc(G) ≤ 2 + 1

k .

• For any triangle-free hexagonal graph G it holds 2 ≤ χc(G) ≤ 5

2 .

• For any triangle-free hexagonal graph G with odd girth 2K + 1 it holds

2K+1

K

≤ χc(G) ≤ 5

2 .

The rest of the paper is organized as follows. In Section 2 some definitions

and results, which will be needed later on, are given. In Section 3 the connection

between graph homomorphisms and circular chromatic number is presented. In

Section 4 the proposition presented in Section 3 is improved and bounds for

circular chromatic number of triangle-free hexagonal graphs are given. In the

last section a conjecture regarding circular chromatic number of triangle-free

hexagonal graphs is set up.

2

2 Preliminaries

Let G and H be simple graphs. It is well known that the existence of a homomorphism

ϕ : G → H implies the inequality χ(G) ≤ χ(H). Namely, for a

homomorphism ψ : H → Kn, the compositum ψ ◦ ϕ : G → Kn is a proper

n-coloring of G.

It is not difficult to see that similar holds for circular chromatic numbers of

graphs G and H.

Lemma 2 If there is a homomorphism f : G → H, then χc(G) ≤ χc(H).

Proof. Let the Euclidean length of the cycle C be equal to r and let c :

V (H) → C be an r-circular coloring of H. Let us show that the compositum

c ◦ f : V (G) → C is an r-circular coloring of G. For any edge xy ∈ E(G)

holds f(x)f(y) ∈ E(H). Since c is an r-circular coloring of H it holds c(f(x)) ∩

c(f(y)) = ∅ for any xy ∈ E(G) and hence c ◦ f is an r-circular coloring of G.

Therefore χc(G) ≤ χc(H).

Let us present another approach to r-circular coloring, which will be needed

in the following section.

The circle C may be cut at an arbitrary point to obtain an interval of length

r, which may be identified with the interval [0, r). For each arc c(x) of C, we let

c ′ (x) be the initial point of c(x) (where c(x) is viewed as going around the circle

C in the clockwise direction). An r-circular coloring of G can be identified with

a mapping c ′ : V → [0, r) such that 1 ≤ |c ′ (x) − c ′ (y)| ≤ r − 1 .

For a later reference we introduce the following definition:

Definition 3 For an arbitrary odd cycle C2k+1 let F : [0, 2k+1

k ) → C2k+1 be a

mapping such that

for x ∈

i i + 1

, ; i ∈ {0, 1, ..., 2k} : F (x) =

k k

0 ; i = 0

2k − 2i + 1 ; 1 ≤ i ≤ k

4k − 2i + 2 ; k < i ≤ 2k

It is not difficult to see that F maps the interval [0, 2 + 1

k ) into vertices

{0, 1, ..., 2k} of the cycle C2k+1 as Figure 1 shows.

Figure 1: The functional values of subintervals [ i

k

of function F defined in Definition 3.

i+1

2k+1

, k ) of the interval [0, k )

Considering Definition 3 and Figure 1 one can easily find out that the following

lemma holds, thus we omit technical details of the proof:

3

.

Lemma 4 Let F : [0, 2 + 1

k ) → C2k+1 be a mapping from Definition 3. For any

vertices x, y ∈ [0, 2 + 1

k ) the following statements are equivalent:

(i) |F (x) − F (y)| = 1,

(ii) 1 − 1

k

< |x − y| < 1 + 2

k .

3 The connection between graph homomorphisms

to odd cycle and circular chromatic number

The Proposition 5 below follows from results given in . For completeness we

give an independent proof of the Proposition in the continuation.

Proposition 5 For any finite graph G there exists a homomorphism f : G →

C2k+1 iff χc(G) ≤ 2k+1 1

k = 2 + k .

Proof. Let f : G → C2k+1 be a homomorphism. Considering Lemma 2 and

the well known equality χc(C2k+1) = 2k+1

k , we have χc(G) ≤ 2k+1

k .

Now suppose χc(G) ≤ 2k+1

k

. Therefore, there exists a 2k+1

k

G, which can be identified with a mapping c ′ : V (G) → [0, 2k+1

k

1 ≤ |c ′ (x) − c ′ (y)| ≤ 1 + 1

k

-circular coloring of

), such that

for every edge xy ∈ E(G). (1)

Let F : [0, 2k+1

k ) → C2k+1 be a mapping from Definition 3. We will prove that

the composition F ◦ c ′ : V (G) → C2k+1 is a homomorphism from G to C2k+1.

Let xy ∈ E(G). We have to show that F (c ′ (x))F (c ′ (y)) ∈ E(C2k+1), which is

equivalent to |F (c ′ (x)) − F (c ′ (y))| = 1. Suppose the opposite (∃x0y0 ∈ E(G)

such that |F (c ′ (x0)) − F (c ′ (y0))| = 1). From Lemma 4 it follows that the assertion

(1 − 1

k < |c′ (x) − c ′ (y)| < 1 + 2

k ) is not true. Hence either |c′ (x) − c ′ (y)| ≤

1 − 1

k < 1 or |c′ (x) − c ′ (y)| ≥ 1 + 2

1

k > 1 + k . Both cases are contradictious to

the inequalities (1). Therefore, |F (c ′ (x)) − F (c ′ (y))| = 1 for every xy ∈ E(G)

or mapping F ◦ c ′ : G → C2k+1 is a homomorphism.

4 Corollaries of Proposition 5

The Proposition 5 can be improved further.

Corollary 6 For an arbitrary graph G the following two statements are equivalent:

(i) k is the biggest positive integer for which there exists a homomorphism

f : G → C2k+1,

(ii) 2 + 1

k+1 < χc(G) ≤ 2 + 1

k .

Proof. (i) ⇒ (ii) : Since f : G → C2k+1 is a homomorphism, by Proposition 5,

. Because there does not exist a homomorphism from G

we have χc(G) ≤ 2 + 1

k

to C2(k+1)+1, the Proposition 5 implies χc(G) > 2(k+1)+1

k+1

4

= 2 + 1

k+1 .

(ii) ⇒ (i) : Because of the inequality χc(G) ≤ 2 + 1

k , by Proposition 5, there

exists a homomorphism f : G → C2k+1. Suppose that there exists a positive

integer n ≥ k + 1 such that there is a homomorphism from G to C2n+1. By

, which is a contradiction.

Proposition 5 we have χc(G) ≤ 2 + 1

n

≤ 2 + 1

k+1

Let G be an arbitrary triangle-free hexagonal graph. It is interesting to ask

what is the circular chromatic number of G. Since an even cycle C2n can be a

subgraph of a hexagonal graph, the well known equality χc(G) = 2 implies the

lower bound i.e. 2 ≤ χc(G). To obtain the upper bound we use the result from

:

Theorem 7 Let G be a triangle-free hexagonal graph. Then there exists a homomorphism

ϕ : G → C5.

Since there exists a homomorphism from G into C5 Proposition 5 implies

. So we proved the following result:

the inequality χc(G) ≤ 5

2

Proposition 8 For any triangle-free hexagonal graph G it holds 2 ≤ χc(G) ≤

5

2 .

Odd girth of a graph G is the length of a shortest odd cycle in G. If there is no

odd cycle, i.e. the graph is bipartite, then the odd girth is undefined. Note that

the smallest odd cycle which can be realized as a triangle free hexagonal graph

is C9. Clearly, for a graph with odd girth 2K + 1, there is no homomorphism

f : G → C 2(K+1)+1, and hence

Proposition 9 For any triangle-free hexagonal graph G with odd girth 2K + 1

it holds 2 + 1

K+1 ≤ χc(G) ≤ 5

2 .

5 Final remarks

In  we conjectured that every triangle-free hexagonal graph is C7-colorable.

If this conjecture is true then it improves the upper bound of Proposition 8.

Therefore, we set another conjecture

Conjecture 10 For any triangle-free hexagonal graph G it holds 2 ≤ χc(G) ≤

7

3 .

References

 J.A. Bondy and P. Hell, A note on the star chromatic number, J.Graph

Theory 14 (1990), 479-482.

 G. Hahn and G. McGillivray, Graph homomorphisms: computational aspects

and infinite graphs, submitted for publication.

5

 G. Hahn and C. Tardif, Graph homomorphisms: structure and symmetry,

in: Graph symmetry, ASI ser.C, Kluwer, 1997, pp.107-166.

 P.Hell and J. Neˇsetril, On the complexity of H-colorings, J.Combin. Theory

B 48 (1990), 92-110.

 P. ˇ Sparl, J. ˇ Zerovnik, Homomorphisms of hexagonal graphs to odd cycles,

Discrete mathematics 283 (2004), 273-277.

 A.Vince, Star chromatic number, J. Graph Theory 12 (1988), 551-559.

 X. Zhu, Circular chromatic number: a survey, Discrete mathematics 229

(2001), 371-410.

 X. Zhu, Circular coloring and graph homomorphism, Bulletin of the Australian

Mathematical Society 59 (1999), 83-97.

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