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

Jadranska 19,

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 sub**graphs** **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

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

was given by Hell and Neˇsetril in 1990 [4]. 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 [3]. One **of** the

restricted H-coloring problems was studied in [5], 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 [8] 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**” [6]. Here we present an equivalent definition **of** Zhu [7].

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** **number**s

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.

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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** **number**s **of**

**graphs** G and H.

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

Pro**of**. 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 [7].

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 pro**of**:

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 [8]. For completeness we

give an independent pro**of** **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 .

Pro**of**. 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 .

Pro**of**. (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

[5]:

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 [5] 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

[1] J.A. Bondy and P. Hell, A note on the star **chromatic** **number**, J.Graph

Theory 14 (1990), 479-482.

[2] G. Hahn and G. McGillivray, Graph homomorphisms: computational aspects

and infinite **graphs**, submitted for publication.

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

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

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

B 48 (1990), 92-110.

[5] P. ˇ Sparl, J. ˇ Zerovnik, Homomorphisms **of** **hexagonal** **graphs** to odd cycles,

Discrete mathematics 283 (2004), 273-277.

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

[7] X. Zhu, **Circular** **chromatic** **number**: a survey, Discrete mathematics 229

(2001), 371-410.

[8] X. Zhu, **Circular** coloring and graph homomorphism, Bulletin **of** the Australian

Mathematical Society 59 (1999), 83-97.

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