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The game coloring number of planar graphs with a given girth
Discrete Mathematics 330 (2014) 11–16
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
The game coloring number of planar graphs with a given girth Yosuke Sekiguchi Department of Mathematics, Keio University, Yokohama 223-8522, Japan
article
abstract
info
Article history: Received 26 July 2012 Received in revised form 27 August 2013 Accepted 14 April 2014 Available online 3 May 2014
This paper discusses the game coloring number of planar graphs. Let G be a planar graph and let colg (G) be the game coloring number of G. We prove that colg (G) is at most 13 if G is a planar graph with girth at least 4. We also show that there is a planar graph G with girth 4 such that colg (G) ≥ 7 and there is a planar graph with girth 5 such that colg (G) ≥ 6. © 2014 Elsevier B.V. All rights reserved.
Keywords: Game coloring number Game chromatic number Planar graph Girth
1. Introduction Given a simple graph G = (V (G), E (G)), the game coloring number of G is defined through a two-person game: the marking game. Two players, say Alice and Bob, with Alice playing first, take turns marking unmarked vertices of G until all vertices are marked. For a vertex v of G, let b(v) be the number of neighbors of v that are marked before v is marked. We define the score s of the game as s := 1 + max b(v). v∈V (G)
Alice’s goal is to minimize the score, while Bob’s goal is to maximize it. The game coloring number colg (G) of G is the least s such that Alice has a strategy that results in a score of at most s. The game coloring number was formally introduced by Zhu [11] as a tool in study of the game chromatic number. The game chromatic number of G is also defined through a two-person game: the chromatic game. Let C be a set of colors. Two players, say Alice and Bob, with Alice playing first, take turns coloring uncolored vertices of G with a color from the color set C so that no two adjacent vertices receive the same color. Alice wins the game if all vertices of G are colored. Otherwise Bob wins the game. The game chromatic number χg (G) of G is the least number of colors so that Alice has a winning strategy. The game chromatic number was introduced by Bodlaender [1]. It is easy to see that χg (G) ≤ colg (G). Moreover, it is easier to deal with the game coloring number than the game chromatic number for several reasons. So, for many classes of graphs, the best known upper bounds for their game chromatic numbers are obtained by finding upper bounds for their game coloring numbers. Suppose H is a family of graphs. We define the game chromatic number and the game coloring number of H as
χg (H ) := max χg (G) : G ∈ H and colg (H ) := max colg (G) : G ∈ H .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2014.04.011 0012-365X/© 2014 Elsevier B.V. All rights reserved.
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Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
We denote by F the family of forests, by P the family of planar graphs, by Ik the family of interval graphs with clique number k, by Q the family of outerplanar graphs, and by P T k the family of partial k-trees. The exact value of the game coloring numbers of F , Ik , Q, and P T k is known. It is proved by Faigle et al. [3] that χg (F ) = colg (F ) = 4, by Faigle et al. [3] and Kierstead and Yang [7] that colg (Ik ) = 3k − 2, by Guan and Zhu [4] and Kierstead and Yang [7] that colg (Q) = 7, and by Wu and Zhu [10] and Zhu [12] that colg (P T k ) = 3k + 2 for k ≥ 2. For P , the best known upper bound is 17 and the best known lower bound is 11 that is 11 ≤ colg (P ) ≤ 17 [10,13]. Some improvements of this result are known. Let Pk be the family of planar graphs with girth at least k. It is proved by He et al. [5] that colg (P5 ) ≤ 8, by Kleitman [8] that colg (P6 ) ≤ 6, by Wang and Zhu [9] that colg (P8 ) ≤ 5, and by Borodin et al. [2] that if G is a quadrangle-free planar graph then colg (G) ≤ 9. One way to show an upper bound for the game coloring number is by considering Alice’s strategy. It is proved in [6] that there is a strategy, called the activation strategy, such that if Alice uses this strategy to play the marking game then she achieves the sharp upper bounds on the game coloring number of F , Ik , Q, and P T k . The activation strategy only depends on a linear ordering on the vertex set of a graph, so we can estimate an upper bound for the game coloring number to construct a linear ordering on it. In this paper, we prove that the game coloring number of planar graphs with girth 4 is at most 13 by constructing an appropriate linear ordering. We also show that there is a planar graph G with girth 4 such that colg (G) ≥ 7 and that there is a planar graph G with girth 5 such that colg (G) ≥ 6. Theorem 1.1. Let Pk be the family of planar graphs with girth at least k. Then the following statements hold. (i) 7 ≤ colg (P4 ) ≤ 13. (ii) 6 ≤ colg (P5 ) ≤ 8. 2. Upper bounds for the game coloring number of planar graphs with girth 4 First, we introduce some definitions and notations. For a graph G, let Π (G) be the set of linear orderings on V (G). The orientation of G with respect to L ∈ Π (G) is defined as a directed graph GL := (V (G), EL ) where v >L u means that v is greater than u with respect to L and EL := {(v, u) : {v, u} ∈ E (G), v >L u}. For a vertex u ∈ V (G), we denote the set of neighbors of u in G by NG (u), the set of out-neighbors of u in GL by NG+L (u), and
− − + the set of in-neighbors of u in GL by NG−L (u). Let dG (u) := |NG (u)|, d+ GL (u) := |NGL (u)|, and dGL (u) := |NGL (u)| be the degree of u, the outdegree of u, and the indegree of u, respectively. We define
VG+L (u) := {v ∈ V (GL ) : v
VG−L (u) := {v ∈ V (GL ) : v >L u}.
Moreover, let NG+L [u] := NG+L (u) ∪ {u},
NG−L [u] := NG−L (u) ∪ {u},
VG+L [u] := VG+L (u) ∪ {u},
VG−L [u] := VG−L (u) ∪ {u}.
When G and GL are clear from the context we will omit the subscripts. Fix a graph G and a linear ordering L on V (G). Then a good strategy for Alice with respect to L is known, the activation strategy [6]. If Alice takes this strategy, then the score of the game is bounded. To state this, we define some parameters. Definition 2.1. Let G be a graph. For subsets A and B of V (G) and a matching M of G, we say that M is a matching from A to B if all edges of M join a vertex of A and a vertex of B \ A and M covers all vertices of A. For u ∈ V (G), the matching number m(u, L, G) of u with respect to L in G is defined to be the size of a largest set Z ⊆ N − [u] such that there is a partition Z = X ∪ Y (X ⊆ N − [u], Y ⊆ N − (u), X ∩ Y = ∅) and there exist a matching M from X to V + (u) and a matching N from Y to V + [u]. Let r (u, L, G) := d+ GL (u) + m(u, L, G), r (L, G) := max r (u, L, G), u∈V (G)
r (G) := min r (L, G). L∈Π (G)
The following theorem holds. Theorem 2.2 (Kierstead [6]). For any graph G and ordering L ∈ Π (G), if Alice uses the activation strategy with respect to L, then the score will be at most 1 + r (L, G). In particular, colg (G) ≤ 1 + r (G).
Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
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For more information on the activation strategy, see [6]. Using Theorem 2.2, we can show that the game coloring number of planar graphs with girth at least 4 is at most 13. Theorem 2.3. If G is a planar graph with girth at least 4, then colg (G) ≤ 13. Hence colg (P4 ) ≤ 13. Proof. It suffices to show that there is a linear ordering L such that r (L, G) ≤ 12. Fix a planar embedding of G. We will construct L as follows. Initially, we have a set of chosen vertices C := ∅, and a set of unchosen vertices U := V (G). At any stage of the construction we choose a vertex u ∈ U, and replace U by U − {u} and C by C ∪ {u}. We arrange the vertices of G so that u
c (v) =
v∈V (H )
c ′ (v) < 6|V (H )|
v∈V (H )
by Euler’s formula. So there exists a vertex u ∈ U such that c ′ (u) ≤ 5.5 (note that any vertex of C satisfies c ′ (v) ≥ 6). We choose such u. Let
σ := |{y ∈ U : {u, y} ∈ S }|, α := |{x ∈ C : {u, x} ∈ A}|,
σ ′ := |{y ∈ U : {u, y} ∈ S ′ }|, β := |{x ∈ C : {u, x} ∈ B}|.
Then c ′ ( u) = σ + σ ′ +
1 2
α + β ≤ 5.5.
We claim that r (u, L, G) ≤ σ + 2σ ′ + α + β + δ + 1, where
δ :=
0 1
if σ = 0, if σ ≥ 1.
′ − Clearly, d+ GL (u) = σ . Thus we will show that m(u, L, G) ≤ 2σ +α +β +δ + 1. Consider a set Z ⊆ N [u] with |Z | = m(u, L, G) − + such that there exists a partition Z = X ∪ Y and there exist matchings M from X ⊆ N [u] to V (u) and N from Y ⊆ N − (u) to V + [u]. Since the girth of G is at least 4, there is no edge joining NG−L (u) and NG+L (u). Hence the number of edges of a matching
from {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≤ 3} to VG+L (u) is at most σ ′ . Therefore
|X \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| ≤ σ ′ + δ, |Y \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| ≤ σ ′ + 1. As |{v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| ≤ α + β we have m(u, L, G) = |Z | = |X | + |Y |
= |X \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| + |Y \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| + |{v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4} ∩ Z | ≤ σ′ + δ + σ′ + 1 + α + β ≤ 2σ ′ + α + β + δ + 1. Hence r (u, L, G) ≤ σ + 2σ ′ + α + β + δ + 1.
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Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
Finally, we show r (u, L, G) ≤ 12. Case 1. σ = 0. In this case, δ = 0. Since c ′ (u) = σ ′ +
1 2
α + β ≤ 5.5
and σ ′ , α , and β are nonnegative integers we have
σ + 2σ ′ + α + β + δ + 1 = 2σ ′ + α + β + 1 ≤ 12. Case 2. σ ≥ 1. In this case, δ = 1. Let σ ∗ := σ − 1. Since c ′ (u) = σ + σ ′ +
1
α+β
2 1
= σ∗ + σ′ + α + β + 1 ≤ 5.5,
2
we have 1
σ ∗ + σ ′ + α + β ≤ 4.5. 2
As σ ∗ , σ ′ , α , and β are nonnegative integers
σ + 2σ ′ + α + β + δ + 1 = σ ∗ + 1 + 2σ ′ + α + β + 1 + 1 = σ ∗ + 2σ ′ + α + β + 3 ≤ 12 holds. Therefore r (u, L, G) ≤ 12 follows. Hence colg (G) ≤ 13.
3. Lower bounds for the game coloring number of planar graphs with given girth In this section, we consider lower bounds for the game coloring number. First, we will show the following lemma. Lemma 3.1. There is a planar graph G with girth 4 such that colg (G) ≥ 7. Proof. We construct G as follows. Let A := ai,j : i, j ∈ {1, 2, . . . , 15} ,
B := bi,j : i ∈ {2, 4, . . . , 14}, j ∈ {1, 2, . . . , 14} ,
C := ci,j : i, j ∈ {1, 2, . . . , 15} ,
EA := {ai,j , ai,j+1 } : ai,j , ai,j+1 ∈ A ∪ {ai,j , ai+1,j } : ai,j , ai+1,j ∈ A ,
EB := {bi,j , ai+1,j } : bi,j ∈ B, ai+1,j ∈ A ∪ {bi,j , ai,j+1 } : bi,j ∈ B, ai,j+1 ∈ A ,
EC := {aij , cij } : i, j ∈ {1, 2, . . . , 15} .
We define G := (A ∪ B ∪ C , EA ∪ EB ∪ EC ) (see Fig. 1). G is a planar graph and its girth is 4. We show that colg (G) ≥ 7. Let B′ := B ∪ {ai,j ∈ A : i ∈ {1, 15} or j ∈ {1, 15}}. First, Bob marks all the vertices of B′ . Since 154 = |B′ | < |A \ B′ | = 169, Bob can mark all the vertices of B′ before all the vertices of A \ B′ are marked. Consider Bob’s turn immediately after all the vertices of B′ are marked. Then there are at least 14 unmarked vertices of A \ B. Let ai1 ,j1 , . . . , aik ,jk be such vertices of A \ B′ where k is the number of unmarked vertices of A \ B′ . Bob can mark ci1 ,j1 , . . . , cik ,jk earlier than ai1 ,j1 , . . . , aik ,jk . Therefore there is a vertex v ∈ A \ B′ which satisfies 1 + b(v) ≥ 7 (note that v has 4 neighbors in A, 1 neighbor in B and 1 neighbor in C ). Hence colg (G) ≥ 7. We also show that the game coloring number of P5 is at least 6. Lemma 3.2. There is a planar graph G with girth 5 such that colg (G) ≥ 6.
Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
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Fig. 1. The graph G which satisfies colg (G) ≥ 7 (we omit the vertices of C and the edges of EC ).
Fig. 2. The construction of graph Hn .
Fig. 3. The graph Gn which satisfies colg (Gn ) ≥ 6 (we omit the vertices of B).
Proof. Let H1 be a regular hexagon (i.e. the cycle of length 6). We construct Hn (n ≥ 2) from Hn−1 by adding 6(n − 1) regular hexagons around Hn−1 (see Fig. 2). f
f
Let the vertices which are incident to a face f of Hn other than the outerplane be labeled v1 , . . . , v6 in the clockwise order f
from the upper left. For each face f of Hn other than the outerplane, we add a vertex af and add edges between af and v1 f
and v4 . Moreover, for each v ∈ V (Hn ), we add a vertex bv and add an edge between bv and v . The resulting graph is Gn (see Fig. 3). Gn is a planar graph and its girth is 5. We will show if n ≥ 7 then colg (Gn ) ≥ 6. Let A := {af : f is a face of Hn other than the outerplane}, B := {bv : v ∈ V (Hn )}, VI := {v ∈ V (Hn ) : v is not incident to the outerplane of Hn }, VO := {v ∈ V (Hn ) : v is incident to the outerplane of Hn }. Then we have
|A| = 3n2 − 3n + 1,
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Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
|VI | = 6n2 − 12n + 6, |VO | = 12n − 6. First, Bob marks all the vertices of A and VO . Since |A| + |VO | + 1 < |VI | for n ≥ 7, Bob can mark all the vertices of A and VO before all the vertices of VI are marked. A vertex of VI has 3 neighbors in V (Hn ), 1 neighbor in A, and 1 neighbor in B. Similarly to the proof of Lemma 3.1, there is a vertex v ∈ VI which satisfies 1 + b(v) ≥ 6. Hence colg (Gn ) ≥ 6. From Theorem 2.3, Lemmas 3.1, 3.2, and the fact that colg (P5 ) ≤ 8 [5], Theorem 1.1 follows. References [1] H.L. Bodlaender, On the complexity of some colouring games, Internat. J. Found. Comput. Sci. 2 (1991) 133–147. [2] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, N.N. Sheikh, Decompositions of quadrangle-free planar graphs, Discuss. Math. Graph Theory 29 (2009) 87–99. [3] U. Faigle, U. Kern, H.A. Kierstead, W.T. Trotter, On the game chromatic number of some classes of graphs, Ars Combin. 35 (1993) 143–150. [4] D. Guan, X. Zhu, The game chromatic number of outerplanar graphs, J. Graph Theory 30 (1999) 67–70. [5] W. He, X. Hou, K. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307–317. [6] H.A. Kierstead, A simple competitive graph coloring algorithm, J. Combin. Theory Ser. B 78 (2000) 57–68. [7] H.A. Kierstead, D. Yang, Very asymmetric marking games, Order 22 (2005) 93–107. [8] D. Kleitman, Partitioning the edges of girth 6 planar graph into those of a forest and those of a set of disjoint paths and cycles, Manuscript, 2006. [9] Y. Wang, Q. Zhang, Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Math. 311 (2011) 844–849. [10] J. Wu, X. Zhu, Lower bounds for the game colouring number of partial k-trees and planar graphs, Discrete Math. 308 (2008) 2637–2642. [11] X. Zhu, The game coloring number of planar graphs, J. Combin. Theory Ser. B 75 (1999) 245–258. [12] X. Zhu, Game colouring number of pseudo partial k-trees, Discrete Math. 215 (2000) 245–262. [13] X. Zhu, Refined activation strategy for the marking game, J. Combin. Theory Ser. B 98 (2008) 1–18.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
The game coloring number of planar graphs with a given girth Yosuke Sekiguchi Department of Mathematics, Keio University, Yokohama 223-8522, Japan
article
abstract
info
Article history: Received 26 July 2012 Received in revised form 27 August 2013 Accepted 14 April 2014 Available online 3 May 2014
This paper discusses the game coloring number of planar graphs. Let G be a planar graph and let colg (G) be the game coloring number of G. We prove that colg (G) is at most 13 if G is a planar graph with girth at least 4. We also show that there is a planar graph G with girth 4 such that colg (G) ≥ 7 and there is a planar graph with girth 5 such that colg (G) ≥ 6. © 2014 Elsevier B.V. All rights reserved.
Keywords: Game coloring number Game chromatic number Planar graph Girth
1. Introduction Given a simple graph G = (V (G), E (G)), the game coloring number of G is defined through a two-person game: the marking game. Two players, say Alice and Bob, with Alice playing first, take turns marking unmarked vertices of G until all vertices are marked. For a vertex v of G, let b(v) be the number of neighbors of v that are marked before v is marked. We define the score s of the game as s := 1 + max b(v). v∈V (G)
Alice’s goal is to minimize the score, while Bob’s goal is to maximize it. The game coloring number colg (G) of G is the least s such that Alice has a strategy that results in a score of at most s. The game coloring number was formally introduced by Zhu [11] as a tool in study of the game chromatic number. The game chromatic number of G is also defined through a two-person game: the chromatic game. Let C be a set of colors. Two players, say Alice and Bob, with Alice playing first, take turns coloring uncolored vertices of G with a color from the color set C so that no two adjacent vertices receive the same color. Alice wins the game if all vertices of G are colored. Otherwise Bob wins the game. The game chromatic number χg (G) of G is the least number of colors so that Alice has a winning strategy. The game chromatic number was introduced by Bodlaender [1]. It is easy to see that χg (G) ≤ colg (G). Moreover, it is easier to deal with the game coloring number than the game chromatic number for several reasons. So, for many classes of graphs, the best known upper bounds for their game chromatic numbers are obtained by finding upper bounds for their game coloring numbers. Suppose H is a family of graphs. We define the game chromatic number and the game coloring number of H as
χg (H ) := max χg (G) : G ∈ H and colg (H ) := max colg (G) : G ∈ H .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2014.04.011 0012-365X/© 2014 Elsevier B.V. All rights reserved.
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Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
We denote by F the family of forests, by P the family of planar graphs, by Ik the family of interval graphs with clique number k, by Q the family of outerplanar graphs, and by P T k the family of partial k-trees. The exact value of the game coloring numbers of F , Ik , Q, and P T k is known. It is proved by Faigle et al. [3] that χg (F ) = colg (F ) = 4, by Faigle et al. [3] and Kierstead and Yang [7] that colg (Ik ) = 3k − 2, by Guan and Zhu [4] and Kierstead and Yang [7] that colg (Q) = 7, and by Wu and Zhu [10] and Zhu [12] that colg (P T k ) = 3k + 2 for k ≥ 2. For P , the best known upper bound is 17 and the best known lower bound is 11 that is 11 ≤ colg (P ) ≤ 17 [10,13]. Some improvements of this result are known. Let Pk be the family of planar graphs with girth at least k. It is proved by He et al. [5] that colg (P5 ) ≤ 8, by Kleitman [8] that colg (P6 ) ≤ 6, by Wang and Zhu [9] that colg (P8 ) ≤ 5, and by Borodin et al. [2] that if G is a quadrangle-free planar graph then colg (G) ≤ 9. One way to show an upper bound for the game coloring number is by considering Alice’s strategy. It is proved in [6] that there is a strategy, called the activation strategy, such that if Alice uses this strategy to play the marking game then she achieves the sharp upper bounds on the game coloring number of F , Ik , Q, and P T k . The activation strategy only depends on a linear ordering on the vertex set of a graph, so we can estimate an upper bound for the game coloring number to construct a linear ordering on it. In this paper, we prove that the game coloring number of planar graphs with girth 4 is at most 13 by constructing an appropriate linear ordering. We also show that there is a planar graph G with girth 4 such that colg (G) ≥ 7 and that there is a planar graph G with girth 5 such that colg (G) ≥ 6. Theorem 1.1. Let Pk be the family of planar graphs with girth at least k. Then the following statements hold. (i) 7 ≤ colg (P4 ) ≤ 13. (ii) 6 ≤ colg (P5 ) ≤ 8. 2. Upper bounds for the game coloring number of planar graphs with girth 4 First, we introduce some definitions and notations. For a graph G, let Π (G) be the set of linear orderings on V (G). The orientation of G with respect to L ∈ Π (G) is defined as a directed graph GL := (V (G), EL ) where v >L u means that v is greater than u with respect to L and EL := {(v, u) : {v, u} ∈ E (G), v >L u}. For a vertex u ∈ V (G), we denote the set of neighbors of u in G by NG (u), the set of out-neighbors of u in GL by NG+L (u), and
− − + the set of in-neighbors of u in GL by NG−L (u). Let dG (u) := |NG (u)|, d+ GL (u) := |NGL (u)|, and dGL (u) := |NGL (u)| be the degree of u, the outdegree of u, and the indegree of u, respectively. We define
VG+L (u) := {v ∈ V (GL ) : v
VG−L (u) := {v ∈ V (GL ) : v >L u}.
Moreover, let NG+L [u] := NG+L (u) ∪ {u},
NG−L [u] := NG−L (u) ∪ {u},
VG+L [u] := VG+L (u) ∪ {u},
VG−L [u] := VG−L (u) ∪ {u}.
When G and GL are clear from the context we will omit the subscripts. Fix a graph G and a linear ordering L on V (G). Then a good strategy for Alice with respect to L is known, the activation strategy [6]. If Alice takes this strategy, then the score of the game is bounded. To state this, we define some parameters. Definition 2.1. Let G be a graph. For subsets A and B of V (G) and a matching M of G, we say that M is a matching from A to B if all edges of M join a vertex of A and a vertex of B \ A and M covers all vertices of A. For u ∈ V (G), the matching number m(u, L, G) of u with respect to L in G is defined to be the size of a largest set Z ⊆ N − [u] such that there is a partition Z = X ∪ Y (X ⊆ N − [u], Y ⊆ N − (u), X ∩ Y = ∅) and there exist a matching M from X to V + (u) and a matching N from Y to V + [u]. Let r (u, L, G) := d+ GL (u) + m(u, L, G), r (L, G) := max r (u, L, G), u∈V (G)
r (G) := min r (L, G). L∈Π (G)
The following theorem holds. Theorem 2.2 (Kierstead [6]). For any graph G and ordering L ∈ Π (G), if Alice uses the activation strategy with respect to L, then the score will be at most 1 + r (L, G). In particular, colg (G) ≤ 1 + r (G).
Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
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For more information on the activation strategy, see [6]. Using Theorem 2.2, we can show that the game coloring number of planar graphs with girth at least 4 is at most 13. Theorem 2.3. If G is a planar graph with girth at least 4, then colg (G) ≤ 13. Hence colg (P4 ) ≤ 13. Proof. It suffices to show that there is a linear ordering L such that r (L, G) ≤ 12. Fix a planar embedding of G. We will construct L as follows. Initially, we have a set of chosen vertices C := ∅, and a set of unchosen vertices U := V (G). At any stage of the construction we choose a vertex u ∈ U, and replace U by U − {u} and C by C ∪ {u}. We arrange the vertices of G so that u
c (v) =
v∈V (H )
c ′ (v) < 6|V (H )|
v∈V (H )
by Euler’s formula. So there exists a vertex u ∈ U such that c ′ (u) ≤ 5.5 (note that any vertex of C satisfies c ′ (v) ≥ 6). We choose such u. Let
σ := |{y ∈ U : {u, y} ∈ S }|, α := |{x ∈ C : {u, x} ∈ A}|,
σ ′ := |{y ∈ U : {u, y} ∈ S ′ }|, β := |{x ∈ C : {u, x} ∈ B}|.
Then c ′ ( u) = σ + σ ′ +
1 2
α + β ≤ 5.5.
We claim that r (u, L, G) ≤ σ + 2σ ′ + α + β + δ + 1, where
δ :=
0 1
if σ = 0, if σ ≥ 1.
′ − Clearly, d+ GL (u) = σ . Thus we will show that m(u, L, G) ≤ 2σ +α +β +δ + 1. Consider a set Z ⊆ N [u] with |Z | = m(u, L, G) − + such that there exists a partition Z = X ∪ Y and there exist matchings M from X ⊆ N [u] to V (u) and N from Y ⊆ N − (u) to V + [u]. Since the girth of G is at least 4, there is no edge joining NG−L (u) and NG+L (u). Hence the number of edges of a matching
from {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≤ 3} to VG+L (u) is at most σ ′ . Therefore
|X \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| ≤ σ ′ + δ, |Y \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| ≤ σ ′ + 1. As |{v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| ≤ α + β we have m(u, L, G) = |Z | = |X | + |Y |
= |X \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| + |Y \ {v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4}| + |{v ∈ NG−L (u) : |NG+L (v) ∩ VG+L [u]| ≥ 4} ∩ Z | ≤ σ′ + δ + σ′ + 1 + α + β ≤ 2σ ′ + α + β + δ + 1. Hence r (u, L, G) ≤ σ + 2σ ′ + α + β + δ + 1.
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Finally, we show r (u, L, G) ≤ 12. Case 1. σ = 0. In this case, δ = 0. Since c ′ (u) = σ ′ +
1 2
α + β ≤ 5.5
and σ ′ , α , and β are nonnegative integers we have
σ + 2σ ′ + α + β + δ + 1 = 2σ ′ + α + β + 1 ≤ 12. Case 2. σ ≥ 1. In this case, δ = 1. Let σ ∗ := σ − 1. Since c ′ (u) = σ + σ ′ +
1
α+β
2 1
= σ∗ + σ′ + α + β + 1 ≤ 5.5,
2
we have 1
σ ∗ + σ ′ + α + β ≤ 4.5. 2
As σ ∗ , σ ′ , α , and β are nonnegative integers
σ + 2σ ′ + α + β + δ + 1 = σ ∗ + 1 + 2σ ′ + α + β + 1 + 1 = σ ∗ + 2σ ′ + α + β + 3 ≤ 12 holds. Therefore r (u, L, G) ≤ 12 follows. Hence colg (G) ≤ 13.
3. Lower bounds for the game coloring number of planar graphs with given girth In this section, we consider lower bounds for the game coloring number. First, we will show the following lemma. Lemma 3.1. There is a planar graph G with girth 4 such that colg (G) ≥ 7. Proof. We construct G as follows. Let A := ai,j : i, j ∈ {1, 2, . . . , 15} ,
B := bi,j : i ∈ {2, 4, . . . , 14}, j ∈ {1, 2, . . . , 14} ,
C := ci,j : i, j ∈ {1, 2, . . . , 15} ,
EA := {ai,j , ai,j+1 } : ai,j , ai,j+1 ∈ A ∪ {ai,j , ai+1,j } : ai,j , ai+1,j ∈ A ,
EB := {bi,j , ai+1,j } : bi,j ∈ B, ai+1,j ∈ A ∪ {bi,j , ai,j+1 } : bi,j ∈ B, ai,j+1 ∈ A ,
EC := {aij , cij } : i, j ∈ {1, 2, . . . , 15} .
We define G := (A ∪ B ∪ C , EA ∪ EB ∪ EC ) (see Fig. 1). G is a planar graph and its girth is 4. We show that colg (G) ≥ 7. Let B′ := B ∪ {ai,j ∈ A : i ∈ {1, 15} or j ∈ {1, 15}}. First, Bob marks all the vertices of B′ . Since 154 = |B′ | < |A \ B′ | = 169, Bob can mark all the vertices of B′ before all the vertices of A \ B′ are marked. Consider Bob’s turn immediately after all the vertices of B′ are marked. Then there are at least 14 unmarked vertices of A \ B. Let ai1 ,j1 , . . . , aik ,jk be such vertices of A \ B′ where k is the number of unmarked vertices of A \ B′ . Bob can mark ci1 ,j1 , . . . , cik ,jk earlier than ai1 ,j1 , . . . , aik ,jk . Therefore there is a vertex v ∈ A \ B′ which satisfies 1 + b(v) ≥ 7 (note that v has 4 neighbors in A, 1 neighbor in B and 1 neighbor in C ). Hence colg (G) ≥ 7. We also show that the game coloring number of P5 is at least 6. Lemma 3.2. There is a planar graph G with girth 5 such that colg (G) ≥ 6.
Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
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Fig. 1. The graph G which satisfies colg (G) ≥ 7 (we omit the vertices of C and the edges of EC ).
Fig. 2. The construction of graph Hn .
Fig. 3. The graph Gn which satisfies colg (Gn ) ≥ 6 (we omit the vertices of B).
Proof. Let H1 be a regular hexagon (i.e. the cycle of length 6). We construct Hn (n ≥ 2) from Hn−1 by adding 6(n − 1) regular hexagons around Hn−1 (see Fig. 2). f
f
Let the vertices which are incident to a face f of Hn other than the outerplane be labeled v1 , . . . , v6 in the clockwise order f
from the upper left. For each face f of Hn other than the outerplane, we add a vertex af and add edges between af and v1 f
and v4 . Moreover, for each v ∈ V (Hn ), we add a vertex bv and add an edge between bv and v . The resulting graph is Gn (see Fig. 3). Gn is a planar graph and its girth is 5. We will show if n ≥ 7 then colg (Gn ) ≥ 6. Let A := {af : f is a face of Hn other than the outerplane}, B := {bv : v ∈ V (Hn )}, VI := {v ∈ V (Hn ) : v is not incident to the outerplane of Hn }, VO := {v ∈ V (Hn ) : v is incident to the outerplane of Hn }. Then we have
|A| = 3n2 − 3n + 1,
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Y. Sekiguchi / Discrete Mathematics 330 (2014) 11–16
|VI | = 6n2 − 12n + 6, |VO | = 12n − 6. First, Bob marks all the vertices of A and VO . Since |A| + |VO | + 1 < |VI | for n ≥ 7, Bob can mark all the vertices of A and VO before all the vertices of VI are marked. A vertex of VI has 3 neighbors in V (Hn ), 1 neighbor in A, and 1 neighbor in B. Similarly to the proof of Lemma 3.1, there is a vertex v ∈ VI which satisfies 1 + b(v) ≥ 6. Hence colg (Gn ) ≥ 6. From Theorem 2.3, Lemmas 3.1, 3.2, and the fact that colg (P5 ) ≤ 8 [5], Theorem 1.1 follows. References [1] H.L. Bodlaender, On the complexity of some colouring games, Internat. J. Found. Comput. Sci. 2 (1991) 133–147. [2] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, N.N. Sheikh, Decompositions of quadrangle-free planar graphs, Discuss. Math. Graph Theory 29 (2009) 87–99. [3] U. Faigle, U. Kern, H.A. Kierstead, W.T. Trotter, On the game chromatic number of some classes of graphs, Ars Combin. 35 (1993) 143–150. [4] D. Guan, X. Zhu, The game chromatic number of outerplanar graphs, J. Graph Theory 30 (1999) 67–70. [5] W. He, X. Hou, K. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307–317. [6] H.A. Kierstead, A simple competitive graph coloring algorithm, J. Combin. Theory Ser. B 78 (2000) 57–68. [7] H.A. Kierstead, D. Yang, Very asymmetric marking games, Order 22 (2005) 93–107. [8] D. Kleitman, Partitioning the edges of girth 6 planar graph into those of a forest and those of a set of disjoint paths and cycles, Manuscript, 2006. [9] Y. Wang, Q. Zhang, Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Math. 311 (2011) 844–849. [10] J. Wu, X. Zhu, Lower bounds for the game colouring number of partial k-trees and planar graphs, Discrete Math. 308 (2008) 2637–2642. [11] X. Zhu, The game coloring number of planar graphs, J. Combin. Theory Ser. B 75 (1999) 245–258. [12] X. Zhu, Game colouring number of pseudo partial k-trees, Discrete Math. 215 (2000) 245–262. [13] X. Zhu, Refined activation strategy for the marking game, J. Combin. Theory Ser. B 98 (2008) 1–18.