Discrete Mathematics 306 (2006) 3321–3326www.elsevier.com/locate/disc

Note

Rainbowness of cubic plane graphsStanislav Jendrol’

Institute of Mathematics, P.J. Šafárik University, Jesenná 5, SK-041 54 Košice, Slovak Republic

Received 2 March 2006; received in revised form 2 June 2006; accepted 25 June 2006Available online 28 August 2006

Abstract

The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graphG using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G weprove that

n

2+ �∗

1 − 1� rb(G)�n − �∗0 + 1,

where �∗0 and �∗

1 denote the independence number and the edge independence number, respectively, of the dual graph G∗ of G. Wealso prove that if the dual graph G∗ of an n-vertex cubic 3-connected plane graph G has a perfect matching then

rb(G) = 34 n.

© 2006 Elsevier B.V. All rights reserved.

MSC: 05C15; 52B10

Keywords: Vertex colouring; Rainbowness; Plane graph; Cubic connected plane graphs

1. Introduction

Colouring vertices of plane graphs under restrictions given by faces has recently attracted much attention, see e.g.[4–7] and references there. One natural problem of this kind is the following Ramsey type problem: let us define therainbowness of a connected plane graph G, rb(G), as the minimum number k such that any surjective colour assignment� : V (G) → {1, 2, . . . , k} involves a face all vertices of which receive distinct colours. Problem is to determine therainbowness of the graph G.

We use the standard terminology according to [1] except for few notation defined throughout. However, we recallsome frequently used terms. We consider finite graphs without loops or multiple edges.

For a plane graph G let �0(G) be the independence number of G and �1(G) be the edge independence number of G.Let G∗ be the dual graph to the plane graph G. Then we let �∗

0(G) = �0(G∗) and �∗

1(G) = �1(G∗).

The rainbowness, rb(T ), of plane triangulations T has been recently studied (under the name looseness) byNegami [6]. He proved that for any triangulation T

�0(T ) + 2�rb(T )�2�0(T ) + 1,

E-mail address: stanislav.jendrol@upjs.sk.

0012-365X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.disc.2006.06.012

3322 S. Jendrol’ / Discrete Mathematics 306 (2006) 3321–3326

where �0(T ) is the independence number of T. Ramamurthi and West [7] observed that the following inequality relatingrb(G) to the independence number �0(G) and the chromatic number �0(G) holds for all plane graphs

rb(G)��0(G) + 2�⌈

n

�0(G)

⌉+ 2,

where n = |V (G)|, the number of vertices of a plane graph G.For an n-vertex plane graph G, the four colour theorem yields rb(G)��n/4�+2. If G is triangles-free, then Grötzsch’s

theorem (see [2,9]) gives rb(G)��n/3� + 2. In [7] Ramamurthi and West showed that the above lower bound is tightfor a fixed n when �(G) = 2, 3 and it is within one of being tight for �(G) = 4. They conjectured the following boundfor triangle-free plane graphs.

Conjecture 1.1. If G is n-vertex triangles-free plane graph, n�4, then rb(G)��n/2� + 2.

Ramamurthi and West proved their conjecture for plane graphs with girth at least six. Jungic et al. [4] answered theconjecture in affirmative. Moreover, they proved for plane graphs G with girth g�5 that the rainbowness rb(G) is atleast �(g − 3)/(g − 2)n − (g − 7)/2(g − 2)� + 1 if g is odd and �(g − 3)/(g − 2)n − (g − 6)/2(g − 2)� + 1 if g iseven. The bounds are tight for all pairs n and g with g�4 and n�5g/2 − 3.

In [3] the authors determined the precise values of the rainbowness for all, except for three, graphs of semiregularpolyhedra.

In the present note we investigate connected cubic plane graphs. For this family of graphs we give better boundsthan those mentioned above. The main result of this note is

Theorem 1.2. Let G be an n-vertex connected cubic plane graph. Let �∗0 and �∗

1 be an independence number and edgeindependence number, respectively, of the dual G∗ of the graph G. Then

n

2+ �∗

1 − 1�rb(G)�n − �∗0 + 1. (1)

In Section 2, we prove the lower bound from Theorem 1.2. Section 3 is devoted to the study of upper bounds on therainbowness. First we prove—in Lemma 3.2—the upper bound from Theorem 1.2. Next we show in Theorem 3.3 thatthis bound can be improved by one for a subfamily of the family of cubic 3-connected plane graphs. Then the idea ofthe proof of Theorem 3.3 is used to obtain two different tight upper bounds on rainbowness for graphs from the familyof n-vertex cubic 3-connected plane graphs and for graphs from the wider family of n-vertex cubic connected graphs.In the final Section 4 we discuss the quality of the bounds in Theorem 1.2 and the upper bound in Theorem 3.3.

2. Lower bound

In this section we prove the lower bound in Theorem 2.1.Let G be a cubic connected plane graph. Let M∗={e∗

1, . . . , e∗d} be a maximum matching in G∗. Clearly �∗

1(G)=d=�∗1.

Every edge e∗i = xy of M∗ is associated in G with a pair of two adjacent faces f (x) and f (y) which share an edge ei

in common.Let M = {e1, e2, . . . , ed} be the set of such defined edges of G. Clearly M is a matching. Let V (M) be the set of

vertices incident with the edges of M (i.e. it is a set of end vertices of edges from M). Evidently |V (M)| = 2d = 2�∗1.

Similarly, let F(M) be the set of faces containing an edge from the set M. Observe that if ei �= ej then the pair offaces incident with the edge ei is disjoint with the pair of faces incident with ej . Hence |F(M)| = 2�∗

1. The followingobservation is easy to see

Observation 1. Each face of G has at most one edge in the set M.

Observation 2. If f1 and f2 are two distinct faces from F(G)−F(M) then f1 and f2 do not share any common edge(and any vertex).

S. Jendrol’ / Discrete Mathematics 306 (2006) 3321–3326 3323

Proof. If f1 and f2 would share an edge h then it could be added to the set M and the edge h∗ = f ∗1 f ∗

2 of G∗corresponding to h could extend maximum matching M∗ of G∗, a contradiction. �

Observation 3. Every face f ∈ F(G) − F(M) contains at least two vertices that are not in V (M).

Proof. Let f be a face that is not in F(M). It is easy to see that no two consecutive vertices of f belong to V (M).Otherwise we have a contradiction with Observation 1. �

The following colouring does not involve any rainbow face. First we find the set M={e1, e2, . . . , ed}, d=�∗1. Next we

find the set of faces F(G)−F(M)={f1, f2, . . . , fm} where m=|F(G)|−2�∗1. We colour vertices of the edge ei with

colour i for any i ∈ {1, . . . , d}. For every face fj , j ∈ {1, . . . , m}, choose two vertices that are not in V (M) and colourthem both with the colour d + j . (Note that m maybe 0 but this does not affect the truth of the proof.) The remainingnot yet coloured vertices are coloured with mutually different colours from the set {d + m + 1, . . . , n − d − m}.

Because each face of G is adjacent with a monochromatic edge or with two vertices of the same colour, G does notcontain any rainbow face. In this colouring we have used the following number of colours

n − d − m = n − �∗1 − |F(G)| + 2�∗

1 = n + �∗1 − |F(G)|. (2)

Because G is a cubic connected plane graph we have 3n=2|E(G)|. Using this fact and the Euler’s formula n−|E(G)|+|F(G)| = 2 we obtain

|F(G)| = n + 4

2. (3)

Substituting for |F(G)| from (2) in (3) we find out that the number of used colours is n/2 + �1 − 2. Hence we haveproved that

rb(G)� n

2+ �1 − 1.

3. Upper bounds

In this section we investigate upper bounds on rainbowness of graphs from several families.

Lemma 3.1. Let G be an n-vertex connected plane graph and let {f1, . . . , fk} be a set of faces of G such that no twoamong them have a common vertex. Then

rb(G)�n − k + 1.

Proof. Let V (fi) be the set of vertices incident with the face fi . Suppose there is a (n− k + 1)-colouring of G that hasno rainbow face. Then there are at most |V (fi)| − 1 colours at fi and at most n − ∑k

i=1 |V (fi)| colours at the verticesoutside of the set

⋃ki=1 V (fi). This means that there are at most

n −k∑

i=1

|V (fi)| +k∑

i=1

(|V (fi)| − 1) = n − k

colours used at the vertices of G; a contradiction. �

Observe that the maximum number of faces in a connected cubic plane graph G that no two among them have avertex in common is �∗

0 = �0(G∗), the independence number of G∗, the dual of G. This observation together with

Lemma 3.1 yield the lower bound in Theorem 1.2.

Lemma 3.2. For any n-vertex connected cubic plane graph G there is

rb(G)�n − �∗0 + 1.

The upper bound in Lemma 3.2 can be improved for 3-connected plane graphs G if �∗0 < |F(G)|/2.

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Theorem 3.3. Let G be an n-vertex cubic 3-connected plane graph with m faces. If �∗0 < m/2 then

rb(G)�n − �∗0.

Moreover, this bound is tight.

Proof. Suppose there is a (n − �∗0)-colouring � of a cubic 3-connected plane graph G that has no rainbow face. Let

V (j) be the set of vertices coloured with colour j and let F(j) be the set of faces that are not rainbow because theycontain at least two vertices from the set V (j). Let us estimate the number of pairs (v, f ) with a vertex v from V (j)

and a face f from F(j).Let |V (j)|= aj and |F(j)|= dj . Each face of F(j) contains at least two vertices from V (j), hence there are at least

2dj such pairs. On the other hand each vertex can be incident with at most three faces of F(j) therefore there are atmost 3aj such pairs. Altogether we have

2dj �3aj . (4)

It is easy to see that if aj = 1 then dj = 0, if aj = 2 then dj �2, and for aj �3 we have from (4) that

dj �⌊

3aj

2

⌋�2(aj − 1).

The number of non-rainbow faces in G is at most

n−�∗0∑

j=1

dj �n−�∗

0∑j=1

2(aj − 1) = 2

n−�∗0∑

j=1

aj − 2(n − �∗0) = 2�∗

0.

Because 2�0 < m there is a rainbow face in G; a contradiction. For tightness of the bound see the last section of thispaper. �

Remark. The 3-connectedness is only used to show that dj �2 if aj = 2. In general we have dj �3 if aj = 2.

Theorem 3.4. Let G be an n-vertex cubic 3-connected plane graph. Then

rb(G)�� 34 n�.

Moreover, the bound is tight.

Proof. Suppose there is a �3n/4�-colouring � of G without any rainbow face. Analogously as in the proof of Theorem3.3 and using Euler’s formula we can show that for r = �3n/4�

n

2+ 2 = |F(G)| =

∣∣∣∣∣r⋃

i=1

F(i)

∣∣∣∣∣ �r∑

i=1

|F(i)| =r∑

i=1

di �2r∑

i=1

(ai − 1) = 2(n − r).

Hence n/2 + 2�2n − 2r , which implies r � 34 n − 1, a contradiction. For tightness of the bound 3

4 n see below. �

Theorem 3.5. Let G be an n-vertex 3-connected plane graph and let G∗ have a perfect matching. Then

rb(G) = 3n

4.

Proof. If G∗ has a perfect matching then �∗1 = |F(G)|/2 and, by (3), �∗

1 = (n + 4)/4. This together with the lowerbound of Theorems 1.2 and 3.4 yields our equality. �

If the requirement on 3-connectedness in Theorem 3.4 is relaxed we can prove the following:

S. Jendrol’ / Discrete Mathematics 306 (2006) 3321–3326 3325

Fig. 1. The graphs showing that the upper bound in Theorem 3.6 is tight.

Theorem 3.6. Let G be an n-vertex cubic connected plane graph. Then

rb(G)�⌊

5n + 2

6

⌋.

Moreover, the bound is tight.

Proof. We follow the idea of the proof of Theorem 3.4 with the number of colours r = �(5n + 2)/6�. We have dj = 0if aj = 1 and 2dj �3aj otherwise, see Remark. This implies dj �3(aj − 1). If we use this instead of dj �2(aj − 1)

in the proof of Theorem 3.4, we obtain our upper bound.The following infinite family of graphs Gk , see Fig. 1, constructed by Schiermeyer [8], shows that the bound is tight.

For every k�1, Gk has order 6k + 4, k = 5k + 2 and no rainbow face. Hence 5k + 2 =�(5(6k + 4)− 4)/6�=�(5(6k +4) + 2)/6� − 1. The numbers at the vertices in Fig. 1 denote the colours of these vertices. �

4. Quality of the bounds

The aim of this section is to discuss the quality of the bounds in Theorem 1.2. We show two infinite families ofgraphs for which the lower bound is attained. We also prove that the upper bound in Theorem 3.3 is sharp.

Consider a d-sided prism Dd . It is a 2d-vertex cubic 3-connected plane graph which is in fact a cartesian productP2×Cd of a path P2 and a cycle Cd . It is easy to see that �∗

0(Dd)=�d/2� and �∗1(Dd)=�(d+1)/2�. In [3] there is proved

that rb(Dd)=�(3d−1)/2� for d �3. Because for the prism Dd there is n/2+�∗1 −1=d+�(d+1)/2�−1=�(3d−1)/2�

the lower bound in Theorem 1.2 is tight.Let the dual graph G∗ of an n-vertex cubic 3-connected plane graph has an almost perfect matching, i.e. let �∗

1(G)=(|F(G)| − 1)/2. In this case n = 4k + 2 for some k�1. Then �1(G

∗) = (n + 2)/4 = k + 1. By the lower bound inTheorem 1.2 we have

rb(G)�2k + 1 + k + 1 − 1 = 3k + 1.

On the other hand, by Theorem 3.3, there is

rb(G)�� 34 n� = � 3

4 (4k + 2)� = �3k + 64�.

Hence rb(G) = 3k + 1. So we have proved that the rainbowness of such graphs equals to the lower bound in Theorem1.2.

It is easy to see that the upper bound in Theorem 1.2 is attained for every cubic plane graph consisting of two 2k-gonalfaces, k�2, and k digons. However, we do not know any example of a cubic connected plane graph G having no digonwith rb(G) = n − �∗

0(G) + 1.For the d-sided prism with d even there is rb(Dd)=n− �∗

0(G)= 2d − d/2 = 3d/2. This means that the upper boundof Theorem 3.3 is sharp.

We believe that the following is true.

3326 S. Jendrol’ / Discrete Mathematics 306 (2006) 3321–3326

Conjecture 4.1. For every n-vertex cubic 3-connected plane graph G there is

rb(G) = n

2+ �∗

1(G) − 1.

Acknowledgements

The author is indebted to both anonymous referees for their valuable comments which helped to improve thepresentation of the paper. This work was supported by Science and Technology Assistance Agency under the contractNo. APVT-20-004104. Support of Slovak VEGA Grant 1/3004/06 is acknowledged as well.

Note added in proof

The author of this paper presented to the Sixth Czech-Slovak International Symposium on Combinatorics, GraphTheory, Algoritmus and Application, Prague, July 10–15, 2006 the construction of an infinte family of connected cubicplane graphs (having no loops or multiple edges) for which the upper bound in Theorem 1.2 is tight.

References

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