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Rainbowness of plane graphs
Electronic Notes in Discrete Mathematics 28 (2007) 125–129 www.elsevier.com/locate/endm
Rainbowness of plane graphs Stanislav JENDROL’ 1 Institute of Mathematics ˇ arik University P. J. Saf´ Jesenn´ a 5, SK-041 54 Koˇsice, Slovak Republic
Abstract The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. We give a survey on recent results concerning ranbowness of plane graphs. Keywords: vertex colouring, rainbowness, plane graph, cubic 3-connected plane graphs
1
Introduction
Colouring vertices of plane graphs under restrictions given by faces has recently attracted much attention, see e.g. [4], [5], [6], [7] and references there. One natural problem of this kind is the following Ramsey type problem: Let us define the rainbowness of a connected plane graph G, rb(G), as the minimum number k such that any surjective colour assignment ϕ : V (G) → {1, 2, . . . , k} involves a face all vertices of which receive distinct colours. Problem is to determine the rainbownes of the graph G. We consider finite graphs without loops or multiple edges. 1
Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.035
126
S. Jendrol’ / Electronic Notes in Discrete Mathematics 28 (2007) 125–129
For a plane graph G let α0 (G) be the independence number of G and α1 (G) be the edge independence number of G. Let G∗ be the dual graph to the plane graph G. Then we let α0∗ (G) = α0 (G∗ ) and α1∗ (G) = α1 (G∗ ). The rainbowness, rb(T ), of plane triangulations T has been recently studied (under the name looseness) by Negami [6]. He proved that for any triangulation T α0 (T ) + 2 ≤ rb(T ) ≤ 2α0 (T ) + 1 , where α0 (T ) is the independence number of T . Ramamurthi and West [7] observed that the following inequality relating rb(G) to the independence number α0 (G) and the chromatic number χ0 (G) holds for all plane graphs n +2, rb(G) ≥ α0 (G) + 2 ≥ χ0 (G)
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) ≥ n4 + 2. If G is triangles-free, then Gr¨otzsch’s theorem (see [1], [8]) gives rb(G) ≥ n3 + 2. In [7] Ramamurthi and West showed that the above lower bound is tight for a fixed n when χ0 (G) = 2, 3 and it is within one of being tight for χ0 (G) = 4. They conjectured the following bound for triangle-free plane graphs. Conjecture 1.1 If G is n-vertex triangles-free plane graph, n ≥ 4, then rb(G) ≥ n2 + 2. Ramamurthi and West proved their conjecture for plane graphs with girth ˇ at least six. Jungi´c, Kr´al’ and Skrekovski [4] answered the conjecture in affirmative. Moreover, they proved for plane graphs G with girth g ≥ 5 g−7 that the rainbowness rb(G) is at least g−3 n − 2(g−2) + 1 if g is odd and g−2 g−3 g−6 g−2 n − 2(g−2) + 1 if g is even. The bounds are tight for all pairs n and g − 3. with g ≥ 4 and n ≥ 5g 2 In [3] the authors determined the precise values of the rainbowness for all, except for three, graphs of semiregular polyhedra.
2
Main results
We investigate connected cubic plane graphs. For this family of graphs we give better bounds than those mentioned above. The main result we have obtained is
S. Jendrol’ / Electronic Notes in Discrete Mathematics 28 (2007) 125–129
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Theorem 2.1 Let G be an n-vertex connected cubic plane graph. Let α0∗ and α1∗ be an independence number and edge independence number, respectively, of the dual G∗ of the graph G. Then n + α1∗ − 1 ≤ rb(G) ≤ n − α0∗ + 1 . (1) 2 Moreover, both bounds are tight. The proof of this theorem, except for the sharpness of the upper bound, can be found in [2]. Here we prove that the upper bound is also sharp.
Figure 1 The following infinite family of graphs Gk in Figure 1 shows that the upper bound is tight. For every k ≥ 0, Gk has order 22+4k, a (16+3k)-colouring and no rainbow face. Its dual graph G∗k has the independence number α0 (G∗k ) = α0∗ (Gk ) = 6 + k. Therefore rb(Gk ) ≥ 17 + 3k. On the other hand rb(Gk ) ≤ n − α0∗ (Gk ) + 1 = (22 + 4k) − (6 + k) + 1 = 17 + 3k and we are done. Recall that the numbers at the vertices in Figure 1 denote the colours of these vertices. 2
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The upper bound in Theorem 2.1 can be improved for 3-connected plane graphs G if α0∗ < |F (G)| . 2 Theorem 2.2 Let G be an n-vertex cubic 3-connected plane graph with m faces. If α0∗ < m2 then rb(G) ≤ n − α0∗ . Moreover, this bound is tight. The proof of the above Theorem 2.2 can be found in [2]. Its idea has been used in the proof of the following result. Theorem 2.3 Let G be an n-vertex cubic 3-connected plane graph. Then 3 n . rb(G) ≤ 4 Moreover, the bound is tight. Recall that a matching of a graph G is called a perfect matching and an almost perfect matching if α1 (G) = |V (G)| and α1 (G) = |V (G)|−1 , respectively. 2 2 We have proved in [2] the next theorem. Theorem 2.4 Let G be an n-vertex 3-connected plane graph and let G∗ have a perfect matching or an almost perfect matching. Then 3n . rb(G) = 4 If the requirement on 3-connectedness in Theorem 2.3 is relaxed we can prove the following: Theorem 2.5 Let G be an n-vertex cubic connected plane graph. Then 5n + 2 . rb(G) ≤ 6 Moreover, the bound is tight. The properties of the non-rainbow ( n2 + α1∗ (G) − 2)-colourings of n-vertex cubic plane graphs G used in the proof of the lower bound in Theorem 2.1 in [2] allows us to believe that the following conjecture is true. Conjecture 2.6 For every n-vertex cubic 3-connected plane graph G there is rb(G) =
n + α1∗ (G) − 1 . 2
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Acknowledgement. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-20-004104. Support of Slovak VEGA Grant 1/3004/06 is acknowledged as well.
References [1] H. Gr¨ otzsch, Dreifarbensatz f¨ ur dreikreisfreie Netze of der Kugel, Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nath. Reihe 8 (1959), 109-120. [2] S. Jendrol’, Rainbowness of cubic plane graphs, Discrete Math. (to appear). ˇ Schr¨ [3] S. Jendrol’ and S. otter, On rainbowness of semiregular polyhedra, (submitted). ˇ [4] V. Jungi´c, D. Kr´ al’ and R. Skrekovski, Coloring of plane graphs with no rainbow faces, Combinatorica 26 (2006), 169-182. [5] D. Kr´ al’, On maximum face-constrained coloring of plane graphs with no short face cycles, Discrete Math. 277 (2004), 301-307. [6] S. Negami, Looseness and independence number of maximal planar graphs, Talk to the Japan Workshop on Graph Theory and Combinatorics 2005, Keio University, Yokohama, June 20-24 (2005), Japan. [7] R. Ramamurthi and D.B. West, Maximum face-constrained coloring of plane graphs, Discrete Math. 274 (2004), 233-240. [8] C. Thomassen, Gr¨ otzsch’s 3-color Theorem, J. Combin. Theorey B 62 (1994), 268-279.
Rainbowness of plane graphs Stanislav JENDROL’ 1 Institute of Mathematics ˇ arik University P. J. Saf´ Jesenn´ a 5, SK-041 54 Koˇsice, Slovak Republic
Abstract The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. We give a survey on recent results concerning ranbowness of plane graphs. Keywords: vertex colouring, rainbowness, plane graph, cubic 3-connected plane graphs
1
Introduction
Colouring vertices of plane graphs under restrictions given by faces has recently attracted much attention, see e.g. [4], [5], [6], [7] and references there. One natural problem of this kind is the following Ramsey type problem: Let us define the rainbowness of a connected plane graph G, rb(G), as the minimum number k such that any surjective colour assignment ϕ : V (G) → {1, 2, . . . , k} involves a face all vertices of which receive distinct colours. Problem is to determine the rainbownes of the graph G. We consider finite graphs without loops or multiple edges. 1
Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.035
126
S. Jendrol’ / Electronic Notes in Discrete Mathematics 28 (2007) 125–129
For a plane graph G let α0 (G) be the independence number of G and α1 (G) be the edge independence number of G. Let G∗ be the dual graph to the plane graph G. Then we let α0∗ (G) = α0 (G∗ ) and α1∗ (G) = α1 (G∗ ). The rainbowness, rb(T ), of plane triangulations T has been recently studied (under the name looseness) by Negami [6]. He proved that for any triangulation T α0 (T ) + 2 ≤ rb(T ) ≤ 2α0 (T ) + 1 , where α0 (T ) is the independence number of T . Ramamurthi and West [7] observed that the following inequality relating rb(G) to the independence number α0 (G) and the chromatic number χ0 (G) holds for all plane graphs n +2, rb(G) ≥ α0 (G) + 2 ≥ χ0 (G)
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) ≥ n4 + 2. If G is triangles-free, then Gr¨otzsch’s theorem (see [1], [8]) gives rb(G) ≥ n3 + 2. In [7] Ramamurthi and West showed that the above lower bound is tight for a fixed n when χ0 (G) = 2, 3 and it is within one of being tight for χ0 (G) = 4. They conjectured the following bound for triangle-free plane graphs. Conjecture 1.1 If G is n-vertex triangles-free plane graph, n ≥ 4, then rb(G) ≥ n2 + 2. Ramamurthi and West proved their conjecture for plane graphs with girth ˇ at least six. Jungi´c, Kr´al’ and Skrekovski [4] answered the conjecture in affirmative. Moreover, they proved for plane graphs G with girth g ≥ 5 g−7 that the rainbowness rb(G) is at least g−3 n − 2(g−2) + 1 if g is odd and g−2 g−3 g−6 g−2 n − 2(g−2) + 1 if g is even. The bounds are tight for all pairs n and g − 3. with g ≥ 4 and n ≥ 5g 2 In [3] the authors determined the precise values of the rainbowness for all, except for three, graphs of semiregular polyhedra.
2
Main results
We investigate connected cubic plane graphs. For this family of graphs we give better bounds than those mentioned above. The main result we have obtained is
S. Jendrol’ / Electronic Notes in Discrete Mathematics 28 (2007) 125–129
127
Theorem 2.1 Let G be an n-vertex connected cubic plane graph. Let α0∗ and α1∗ be an independence number and edge independence number, respectively, of the dual G∗ of the graph G. Then n + α1∗ − 1 ≤ rb(G) ≤ n − α0∗ + 1 . (1) 2 Moreover, both bounds are tight. The proof of this theorem, except for the sharpness of the upper bound, can be found in [2]. Here we prove that the upper bound is also sharp.
Figure 1 The following infinite family of graphs Gk in Figure 1 shows that the upper bound is tight. For every k ≥ 0, Gk has order 22+4k, a (16+3k)-colouring and no rainbow face. Its dual graph G∗k has the independence number α0 (G∗k ) = α0∗ (Gk ) = 6 + k. Therefore rb(Gk ) ≥ 17 + 3k. On the other hand rb(Gk ) ≤ n − α0∗ (Gk ) + 1 = (22 + 4k) − (6 + k) + 1 = 17 + 3k and we are done. Recall that the numbers at the vertices in Figure 1 denote the colours of these vertices. 2
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S. Jendrol’ / Electronic Notes in Discrete Mathematics 28 (2007) 125–129
The upper bound in Theorem 2.1 can be improved for 3-connected plane graphs G if α0∗ < |F (G)| . 2 Theorem 2.2 Let G be an n-vertex cubic 3-connected plane graph with m faces. If α0∗ < m2 then rb(G) ≤ n − α0∗ . Moreover, this bound is tight. The proof of the above Theorem 2.2 can be found in [2]. Its idea has been used in the proof of the following result. Theorem 2.3 Let G be an n-vertex cubic 3-connected plane graph. Then 3 n . rb(G) ≤ 4 Moreover, the bound is tight. Recall that a matching of a graph G is called a perfect matching and an almost perfect matching if α1 (G) = |V (G)| and α1 (G) = |V (G)|−1 , respectively. 2 2 We have proved in [2] the next theorem. Theorem 2.4 Let G be an n-vertex 3-connected plane graph and let G∗ have a perfect matching or an almost perfect matching. Then 3n . rb(G) = 4 If the requirement on 3-connectedness in Theorem 2.3 is relaxed we can prove the following: Theorem 2.5 Let G be an n-vertex cubic connected plane graph. Then 5n + 2 . rb(G) ≤ 6 Moreover, the bound is tight. The properties of the non-rainbow ( n2 + α1∗ (G) − 2)-colourings of n-vertex cubic plane graphs G used in the proof of the lower bound in Theorem 2.1 in [2] allows us to believe that the following conjecture is true. Conjecture 2.6 For every n-vertex cubic 3-connected plane graph G there is rb(G) =
n + α1∗ (G) − 1 . 2
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Acknowledgement. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-20-004104. Support of Slovak VEGA Grant 1/3004/06 is acknowledged as well.
References [1] H. Gr¨ otzsch, Dreifarbensatz f¨ ur dreikreisfreie Netze of der Kugel, Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nath. Reihe 8 (1959), 109-120. [2] S. Jendrol’, Rainbowness of cubic plane graphs, Discrete Math. (to appear). ˇ Schr¨ [3] S. Jendrol’ and S. otter, On rainbowness of semiregular polyhedra, (submitted). ˇ [4] V. Jungi´c, D. Kr´ al’ and R. Skrekovski, Coloring of plane graphs with no rainbow faces, Combinatorica 26 (2006), 169-182. [5] D. Kr´ al’, On maximum face-constrained coloring of plane graphs with no short face cycles, Discrete Math. 277 (2004), 301-307. [6] S. Negami, Looseness and independence number of maximal planar graphs, Talk to the Japan Workshop on Graph Theory and Combinatorics 2005, Keio University, Yokohama, June 20-24 (2005), Japan. [7] R. Ramamurthi and D.B. West, Maximum face-constrained coloring of plane graphs, Discrete Math. 274 (2004), 233-240. [8] C. Thomassen, Gr¨ otzsch’s 3-color Theorem, J. Combin. Theorey B 62 (1994), 268-279.