- Email: [email protected]
Decomposition of sparse graphs into two forests, one having bounded maximum degree
Information Processing Letters 110 (2010) 913–916
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Decomposition of sparse graphs into two forests, one having bounded maximum degree Mickael Montassier a,∗,1 , André Raspaud a,1 , Xuding Zhu b,c,2 a b c
Université Bordeaux 1, LaBRI UMR CNRS 5800, France Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan National Center for Theoretical Sciences, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 March 2010 Received in revised form 18 May 2010 Accepted 10 July 2010 Available online 17 July 2010 Communicated by B. Doerr Keywords: Combinatorial problems Decomposition Forest with bounded degree Discharging procedure Global rules
Let G be a graph. The maximum average degree of G, written Mad(G ), is the largest average degree among the subgraphs of G. It was proved in Montassier et al. (2010) [11] that, for any integer k 0, every simple graph with maximum average degree less than mk = 4(k+1)(k+3) admits an edge-partition into a forest and a subgraph with maximum degree at k2 +6k+6 most k; furthermore, when k 3 both subgraphs can be required to be forests. In this note, we extend this result proving that, for k = 4, 5, every simple graph with maximum average degree less than mk admits an edge-partition into two forests, one having maximum degree at most k (i.e. every graph with maximum average degree less than 70 (resp. 192 ) admits 23 61 an edge-partition into two forests, one having maximum degree at most 4 (resp. 5)). © 2010 Elsevier B.V. All rights reserved.
1. Introduction All considered graphs are simple. A decomposition of a graph G is a set of edge-disjoint subgraphs whose union is G. By the well-known Nash-Williams’ Theorem [12], every planar graph decomposes into three forests. Balogh et al. [1] conjectured that one of the three forests can be required to have maximum degree at most 4, which is sharp infinitely often. They proved several results in this direction, and Gonçalves [7] proved the full conjecture. In addition, he showed that planar graphs with girth at least 6 (at least 7) decompose into two forests with one having maximum degree at most 4 (at most 2). From his proof, one can easily derive that every graph with maximum average degree less than 3 decomposes into two forests, one having maximum degree at most 4. We recall that the
*
Corresponding author. E-mail addresses: [email protected] (M. Montassier), [email protected] (A. Raspaud), [email protected] (X. Zhu). 1 Research supported by French–Taiwanese project CNRS/NSC – New trends in graph colorings (2008–2009) and the ANR project GRATEL. 2 Research supported by NSC97-2115-M-110-008-MY3. 0020-0190/$ – see front matter doi:10.1016/j.ipl.2010.07.009
© 2010
Elsevier B.V. All rights reserved.
maximum average degree of a graph G, written Mad(G ), is the largest average degree among the subgraphs of G. Previously, Borodin et al. [6], He et al. [9], and Kleitman [10] were interested by decomposing planar graphs (with large girth) into a forest and a subgraph with bounded maximum degree: every planar graph with girth at least 9 (resp. 6, 5) decomposes into a forest and a subgraph with maximum degree at most 1 [6] (resp. 2 [10], 4 [9]). Such decompositions have applications for the game coloring number; see [13]. In this perspective, decompositions of a graph with given maximum average degree into a forest and a subgraph with bounded maximum degree were stated in [11]. More precisely it was proved that, for any integer k 0, every simple graph with maximum average 4(k+1)(k+3) degree less than mk = k2 +6k+6 decomposes into a forest and a subgraph with maximum degree at most k; furthermore, when k 3 both subgraphs can be required to be forests. This note is a follow up of this work. We prove that: Theorem 1. For k = 4, 5, every simple graph with maximum )(k+3) average degree less than mk = 4(kk2++16k decomposes into two +6 forests, one having maximum degree at most k.
914
M. Montassier et al. / Information Processing Letters 110 (2010) 913–916
In other words, (T1) every simple graph with maximum average degree less 70 than 23 decomposes into two forests, one having maximum degree at most 4, and (T2) every simple graph with maximum average degree less than 192 decomposes into two forests, one having maxi61 mum degree at most 5. Since every planar graph with girth at least g has maximum average degree less than 2g /( g − 2), (T1) implies the result of Gonçalves [7] on planar graphs with girth 6. We do not think that this result is sharp, but the value mk is the largest value our approach can prove (see [11] for more details). The proof of Theorem 1 is based on the method of reducible configurations and on a discharging procedure. This method is classical for this kind of problem. However a distinctive feature of our discharging method is that charge can be transferred to unlimited distances using some key structures called “banks” and “cores”. Idea of “global discharging method” is relatively recent: some applications of this approach can be found in [2–5,8,11]. Section 2 is dedicated to the proof of Theorem 1. 2. Proof of Theorem 1 Let G be a simple graph. Let d(x) denote the degree of x in G. A vertex of degree j (resp. at least j, at most j) is called a j-vertex (resp. j + -vertex, j − -vertex). An (a, b)-alternating cycle is an even cycle that alternates between a-vertices and b-vertices. A jl -vertex is a vertex of degree j adjacent to exactly l 2-vertices. 2.1. Graphs with Mad <
70 23
Suppose that (T1) is false. Let G be a graph with maximum average degree less than 70 that does not admit 23 an edge-partition into two forests, one having maximum degree at most 4. Moreover assume that G minimizes σ (G ) = | V (G )| + | E (G )|. Lemma 1. (See [11].) The counterexample G does not contain (a) 1-vertices, (b) a 2-vertex adjacent to a 5− -vertex, (c) (6, 2)alternating cycles. We give here the proof of Lemma 1 for sake of completeness: Proof. When G contains such a configuration, we decompose an appropriate subgraph of G into two forests F ∞ and F 4 where F 4 has maximum degree at most 4. (a) Let v be a 1-vertex in G, with u the neighbor of v. The decomposition of G − v into F ∞ and F 4 extends to G by adding the edge uv to F ∞ , a contradiction. (b) Let u be a 2-vertex adjacent to a 5− -vertex v. Let w be the other neighbor of u. Consider the decomposition of G − u into F ∞ and F 4 . We add u w in F ∞ . If all the edges incident to v (in G − u) are in F 4 , we add uv in F ∞ and in F 4 otherwise. This extends the decomposition to G, a contradiction.
(c) Let C be a (6, 2)-alternating cycle in G. Consider the decomposition of G − E (C ) into F ∞ and F 4 . We may assume that each 6-vertex on C has an incident edge in F ∞ , since otherwise we can move an incident edge from F 4 to F ∞ . Now, adding one perfect matching in C to F ∞ and the other to F 4 extends the decomposition to G, a contradiction. 2 Let G 3 be the subgraph of G induced by the 3-vertices of G. Lemma 2. Every component of G 3 is either a cycle or a tree. Proof. Let H be a component of G 3 . Suppose that H is not a tree and not a cycle. Let C = x1 . . . xl x1 be a shortest cycle of H and let u be the neighbor of x1 in H not in C . The graph G − ( E (C ) ∪ {x1 u }) decomposes into F ∞ and F 4 . We may assume that each 3-vertex on C − x1 has its incident edge (in G − ( E (C ) ∪ {x1 u })) in F ∞ , since otherwise we can move this incident edge from F 4 to F ∞ . Now adding x1 x2 in F ∞ and all the remaining edges in F 4 extends the decomposition to G, a contradiction. 2 We use now a discharging method to show that every graph satisfying (a), (b), and (c) of Lemma 1 and Lemma 2 70 has average degree at least 23 , and hence there is no counterexample to (T1). To apply the discharging method, we first give each vertex a “charge” equal to its degree. We then redistribute the charge (without changing the total 70 charge) to obtain charge at least 23 on each vertex. To facilitate the discharging argument, we also move some charge to special subgraphs, called banks and cores (defined below). They start with charge 0 and will end with non-negative charge, so the initial average degree is at least 70 . 23 Given such a graph G, let X be the set of all 6-vertices in G that are adjacent to exactly six vertices of degree 2, and let Y be the set of all 2-vertices adjacent to at least one vertex of X . Define the bank of type 1 of G to be the maximal bipartite subgraph of G with partite sets X and Y . A cycle in the bank of type 1 would be a (6, 2)-alternating cycle in G, which is forbidden by Lemma 1(c). Hence the bank of type 1 is a forest. We call each component of the bank of type 1 a core of type 1. By construction, each vertex of X has six neighbors in the bank; hence each leaf in the bank belongs to Y . Define the bank of type 2 of G to be the maximal subgraph of G consisted of 3-vertices. We call each component of the bank of type 2 a core of type 2. By Lemma 2, each core of type 2 is either a cycle or a tree. A vertex belonging to no core is said to be adjacent to a core C if it is adjacent in G to a vertex of C . The discharging rules are defined as follows: R1. Every 6+ -vertex gives 12 to each neighbor that is a 23 2-vertex. R2. If C is a core of type 1 or 2, v is a 4+ -vertex not belonging to a core, and v is adjacent to l vertices of C , 1 to C . then v gives l · 23 R3. Every core of type 1 gives vertices.
4 23
to each of its own 6-
M. Montassier et al. / Information Processing Letters 110 (2010) 913–916
R4. Every core of type 2 gives vertices.
1 23
to each of its own 3-
Observe that a 4+ -vertex may give charge by R1 and R2 (for example, if it is adjacent to 2-vertices and to cores). We first check that, for every vertex v, the new charge 70 ω∗ ( v ) of v after the discharging procedure is at least 23 . By Lemma 1(a), every vertex v has degree at least 2. Suppose that v is a vertex of degree 2. Vertex v starts with charge 2 and receives 12 from each adjacent 6+ 23 vertex by Lemma 1(b) and R1. Hence its new charge is ω∗ ( v ) = 2 + 2 × 12 = 70 . 23 23 Suppose that v is a vertex of degree 3. Then, by definition, it belongs to a core of type 2, say C , and so it receives 1 1 from C by R4. It follows that ω∗ ( v ) = 3 + 23 = 70 . 23 23 Suppose that v is a vertex of degree 4 or 5. Vertex v may be adjacent to cores of type 2. It follows that ω∗ ( v ) 1 d( v ) − d( v ) · 23 = d( v ) · 22 70 by R2. 23 23 Suppose that v is a vertex of degree 6. Assume first that v does not belong to a core. Then v is adjacent to at most five 2-vertices. It follows by R1 and R2, that ω∗ ( v ) 6 − 1 1 5 · 12 − 5 · 23 − 23 = 72 . Assume that v belongs to a core of 23 23 type 1, say C . Then v gives six times 4 from 23 70 . 23
C by R3. It follows that
12 23
by R1 and receives
ω∗ ( v ) = 6 − 6 ·
12 23
+
4 23
=
Suppose finally that v is a vertex of degree at least 7. 1 10 − d( v ) · 23 = d( v ) · 23 70 We have ω∗ ( v ) d( v ) − d( v ) · 12 23 23 by R1 and R2. It remains to prove that the new charge of each core is non-negative. Let C be a core of type 1. Recall that C is a tree consisted of vertices of degree 2 and 6. Let n66 (C ) and nl (C ) be the number of 66 -vertices and leaves of C respectively. Observe that:
nl (C ) 4 · n66 (C )
(1)
1 By R2, C receives nl (C ) · 23 , and gives
4 to each of its own 23 4 66 -vertices by R3 (i.e. it gives n66 (C ) · 23 ). Hence, ∗ (C ) = 1 4 1 4 nl (C ) · 23 − n66 (C ) · 23 4 · n66 (C ) · 23 − n66 (C ) · 23 = 0 by
ω
Eq. (1). Let C be a core of type 2. By Lemma 2, C is a tree or a cycle. Hence the number of edges linking C to G \ C is greater or equal to the number of vertices of C . It follows by R2 and R4 that ω∗ (C ) 0. This completes the proof of (T1). 2.2. Graphs with Mad <
192 61
The proof of (T2) is similar to the proof of (T1). We just give some parts of the proof. Assume that (T2) is false. Let G be a graph with maximum average degree less than 192 61 that does not admit an edge-partition into two forests, one having maximum degree at most 5. Moreover assume that G minimizes σ (G ) = | V (G )| + | E (G )|. By minimality of G, we have: Lemma 3. The counterexample G does not contain 1-vertices, a 2-vertex adjacent to a 6− -vertex, (7, 2)-alternating cycles; moreover each component of G 3 (the graph induced by the 3vertices) is a tree or a cycle.
915
Define now the banks and the cores. Given such a graph G, let X be the set of all 7-vertices in G that are adjacent to exactly seven vertices of degree 2, and let Y be the set of all 2-vertices adjacent to at least one vertex of X . Define the bank of type 1 of G to be the maximal bipartite subgraph of G with partite sets X and Y . A cycle in the bank of type 1 would be a (7, 2)-alternating cycle in G, which is forbidden (by Lemma 3). Hence the bank of type 1 is a forest. We call each component of the bank of type 1 a core of type 1. By construction, each vertex of X has seven neighbors in the bank; hence each leaf in the bank belongs to Y . The bank of type 2 is identical as one of the proof of (T1): define the bank of type 2 of G to be the maximal subgraph of G consisted of 3-vertices. We call each component of the bank of type 2 a core of type 2. Each core of type 2 is either a cycle or a tree. (which To show that G has average degree at least 192 61 is a contradiction), we assign to each vertex a charge equal to its degree and to each core a charge equal to 0; we then apply the following discharging rules: to each neighbor that is a R1. Every 7+ -vertex gives 35 61 2-vertex. R2. If C is a core of type 1, v is a 7+ -vertex belonging to no core, and v is adjacent to l vertices of C , then v 2 to C . gives l · 61 If C is a core of type 2, v is a 4+ -vertex belonging to no core, and v is adjacent to l vertices of C , then v 9 gives l · 61 to C . R3. Every core of type 1 gives tices. R4. Every core of type 2 gives tices.
10 61
to each of its own 7-ver-
9 61
to each of its own 3-ver-
Vertices of G have degree at least 2 by Lemma 3. A vertex v of degree 2 is adjacent to two 7+ -vertices and so by R1; it follows that ω∗ ( v ) = receives two times 35 61 2+2·
35 61
=
192 . 61
A vertex v of degree 3 belongs to a
core C of type 2 by definition, hence v receives C by R4 and ω∗ ( v )
=3+
9 61
=
192 . 61
9 61
from
A vertex v of degree k = 4, 5, 6 can give charges only by R2; it follows that 9 ω∗ ( v ) k − k · 61 208 . Now let v be a vertex of degree 7. 61 Assume first that v does not belong to a core. Then v is adjacent to at most six 2-vertices. It follows by R1 and R2, 2 9 − 6 · 61 − 61 = 196 . Assume that that ω∗ ( v ) 7 − 6 · 35 61 61 v belongs to a core of type 1, say C . Then v gives seven times 35 by R1 and receives 10 from C by R3. It follows 61 61 that
ω∗ ( v ) = 7 − 7 ·
35 61
+
10 61
=
192 . 61
Finally a vertex v of
degree k 8 may give at most k times that ω∗ ( v )
k−k·
35 61
+
2 61
=
24 61
35 61
·k
2 . It follows 61 192 . It remains 61
+
to see that each core has a non-negative final charge. Let C be a core of type 1. Recall that C is a tree consisted of vertices of degree 2 and 7. Let n77 (C ) and nl (C ) be the number of 77 -vertices and leaves of C respectively. Ob2 , serve that: nl (C ) 5 · n77 (C ). By R2, C receives nl (C ) · 61 and gives
10 61
gives n77 (C ) · 2 61
to each of its own 77 -vertices by R3 (i.e. it 10 ). 23
Hence, 10 61
2 ω∗ (C ) = nl (C ) · 61 − n77 (C ) · 10 61
5 · n 77 ( C ) · − n77 (C ) · = 0. Let C be a core of type 2; C is a tree or a cycle. Hence the number of edges linking C
916
M. Montassier et al. / Information Processing Letters 110 (2010) 913–916
to G \ C is greater or equal to the number of vertices of C . It follows by R2 and R4 that ω∗ (C ) 0. This completes the proof of (T2). This approach does not work for higher values of k. In fact, a 4-vertex may give to much charge to 3-vertices. For example, when k = 6, we have mk = 42 . A 3-vertex 13 needs
3 . A 4-vertex v may 13 3 = 40 on v which is 13 13
give 4 times
3 . 13
It remains
4−4· less than mk . A possible way to extend this result for k 6 may be to define a new bank of type 2 including vertices of degree 3 and 4. References [1] J. Balogh, M. Kochol, A. Pluhár, X. Yu, Covering planar graphs with forests, J. Combin. Theory Ser. B 94 (2005) 147–158. [2] O.V. Borodin, S.G. Hartke, A.O. Ivanova, A.V. Kostochka, D.B. West, (5, 2)-Coloring of sparse graphs, Sib. Elektron. Mat. Izv. 5 (2008) 417–426, http://semr.math.nsc.ru. [3] O.V. Borodin, A.O. Ivanova, Near-proper list vertex 2-colorings of sparse graphs, Diskretn. Anal. Issled. Oper. 16 (2) (2009) 16–20 (in Russian). [4] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Oriented vertex 5-coloring of sparse graphs, Diskretn. Anal. Issled. Oper. 13 (1) (2006) 16–32 (in Russian).
[5] O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, A. Raspaud, Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k, J. Graph Theory (2009), doi:10.1002/jgt.20467, in press. [6] O.V. Borodin, A.V. Kostochka, N.N. Sheikh, G. Yu, Decomposing a planar graph with girth 9 into a forest and a matching, European J. Combin. 29 (5) (2008) 1235–1241. [7] D. Gonçalves, Covering planar graphs with forests, one having bounded maximum degree, J. Combin. Theory Ser. B 99 (2009) 314– 322. [8] F. Havet, J.-S. Sereni, Improper choosability of graphs and maximum average degree, J. Graph Theory 52 (2006) 181–199. [9] W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307–317. [10] D. Kleitman, Partitioning the edges of a girth 6 planar graph into those of a forest and those of a set of disjoint paths and cycles, Manuscript, 2006. [11] M. Montassier, A. Pêcher, A. Raspaud, D. West, X. Zhu, Decomposition of sparse graphs, with application to game coloring number, Discrete Math. 310 (10–11) (2010) 1520–1523. [12] C.St.J.A. Nash-Williams, Decompositions of finite graphs into forests, J. London Math. Soc. 39 (12) (1964). [13] X. Zhu, The game coloring number of planar graphs, J. Combin. Theory Ser. B 75 (1999) 245–258.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Decomposition of sparse graphs into two forests, one having bounded maximum degree Mickael Montassier a,∗,1 , André Raspaud a,1 , Xuding Zhu b,c,2 a b c
Université Bordeaux 1, LaBRI UMR CNRS 5800, France Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan National Center for Theoretical Sciences, Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 March 2010 Received in revised form 18 May 2010 Accepted 10 July 2010 Available online 17 July 2010 Communicated by B. Doerr Keywords: Combinatorial problems Decomposition Forest with bounded degree Discharging procedure Global rules
Let G be a graph. The maximum average degree of G, written Mad(G ), is the largest average degree among the subgraphs of G. It was proved in Montassier et al. (2010) [11] that, for any integer k 0, every simple graph with maximum average degree less than mk = 4(k+1)(k+3) admits an edge-partition into a forest and a subgraph with maximum degree at k2 +6k+6 most k; furthermore, when k 3 both subgraphs can be required to be forests. In this note, we extend this result proving that, for k = 4, 5, every simple graph with maximum average degree less than mk admits an edge-partition into two forests, one having maximum degree at most k (i.e. every graph with maximum average degree less than 70 (resp. 192 ) admits 23 61 an edge-partition into two forests, one having maximum degree at most 4 (resp. 5)). © 2010 Elsevier B.V. All rights reserved.
1. Introduction All considered graphs are simple. A decomposition of a graph G is a set of edge-disjoint subgraphs whose union is G. By the well-known Nash-Williams’ Theorem [12], every planar graph decomposes into three forests. Balogh et al. [1] conjectured that one of the three forests can be required to have maximum degree at most 4, which is sharp infinitely often. They proved several results in this direction, and Gonçalves [7] proved the full conjecture. In addition, he showed that planar graphs with girth at least 6 (at least 7) decompose into two forests with one having maximum degree at most 4 (at most 2). From his proof, one can easily derive that every graph with maximum average degree less than 3 decomposes into two forests, one having maximum degree at most 4. We recall that the
*
Corresponding author. E-mail addresses: [email protected] (M. Montassier), [email protected] (A. Raspaud), [email protected] (X. Zhu). 1 Research supported by French–Taiwanese project CNRS/NSC – New trends in graph colorings (2008–2009) and the ANR project GRATEL. 2 Research supported by NSC97-2115-M-110-008-MY3. 0020-0190/$ – see front matter doi:10.1016/j.ipl.2010.07.009
© 2010
Elsevier B.V. All rights reserved.
maximum average degree of a graph G, written Mad(G ), is the largest average degree among the subgraphs of G. Previously, Borodin et al. [6], He et al. [9], and Kleitman [10] were interested by decomposing planar graphs (with large girth) into a forest and a subgraph with bounded maximum degree: every planar graph with girth at least 9 (resp. 6, 5) decomposes into a forest and a subgraph with maximum degree at most 1 [6] (resp. 2 [10], 4 [9]). Such decompositions have applications for the game coloring number; see [13]. In this perspective, decompositions of a graph with given maximum average degree into a forest and a subgraph with bounded maximum degree were stated in [11]. More precisely it was proved that, for any integer k 0, every simple graph with maximum average 4(k+1)(k+3) degree less than mk = k2 +6k+6 decomposes into a forest and a subgraph with maximum degree at most k; furthermore, when k 3 both subgraphs can be required to be forests. This note is a follow up of this work. We prove that: Theorem 1. For k = 4, 5, every simple graph with maximum )(k+3) average degree less than mk = 4(kk2++16k decomposes into two +6 forests, one having maximum degree at most k.
914
M. Montassier et al. / Information Processing Letters 110 (2010) 913–916
In other words, (T1) every simple graph with maximum average degree less 70 than 23 decomposes into two forests, one having maximum degree at most 4, and (T2) every simple graph with maximum average degree less than 192 decomposes into two forests, one having maxi61 mum degree at most 5. Since every planar graph with girth at least g has maximum average degree less than 2g /( g − 2), (T1) implies the result of Gonçalves [7] on planar graphs with girth 6. We do not think that this result is sharp, but the value mk is the largest value our approach can prove (see [11] for more details). The proof of Theorem 1 is based on the method of reducible configurations and on a discharging procedure. This method is classical for this kind of problem. However a distinctive feature of our discharging method is that charge can be transferred to unlimited distances using some key structures called “banks” and “cores”. Idea of “global discharging method” is relatively recent: some applications of this approach can be found in [2–5,8,11]. Section 2 is dedicated to the proof of Theorem 1. 2. Proof of Theorem 1 Let G be a simple graph. Let d(x) denote the degree of x in G. A vertex of degree j (resp. at least j, at most j) is called a j-vertex (resp. j + -vertex, j − -vertex). An (a, b)-alternating cycle is an even cycle that alternates between a-vertices and b-vertices. A jl -vertex is a vertex of degree j adjacent to exactly l 2-vertices. 2.1. Graphs with Mad <
70 23
Suppose that (T1) is false. Let G be a graph with maximum average degree less than 70 that does not admit 23 an edge-partition into two forests, one having maximum degree at most 4. Moreover assume that G minimizes σ (G ) = | V (G )| + | E (G )|. Lemma 1. (See [11].) The counterexample G does not contain (a) 1-vertices, (b) a 2-vertex adjacent to a 5− -vertex, (c) (6, 2)alternating cycles. We give here the proof of Lemma 1 for sake of completeness: Proof. When G contains such a configuration, we decompose an appropriate subgraph of G into two forests F ∞ and F 4 where F 4 has maximum degree at most 4. (a) Let v be a 1-vertex in G, with u the neighbor of v. The decomposition of G − v into F ∞ and F 4 extends to G by adding the edge uv to F ∞ , a contradiction. (b) Let u be a 2-vertex adjacent to a 5− -vertex v. Let w be the other neighbor of u. Consider the decomposition of G − u into F ∞ and F 4 . We add u w in F ∞ . If all the edges incident to v (in G − u) are in F 4 , we add uv in F ∞ and in F 4 otherwise. This extends the decomposition to G, a contradiction.
(c) Let C be a (6, 2)-alternating cycle in G. Consider the decomposition of G − E (C ) into F ∞ and F 4 . We may assume that each 6-vertex on C has an incident edge in F ∞ , since otherwise we can move an incident edge from F 4 to F ∞ . Now, adding one perfect matching in C to F ∞ and the other to F 4 extends the decomposition to G, a contradiction. 2 Let G 3 be the subgraph of G induced by the 3-vertices of G. Lemma 2. Every component of G 3 is either a cycle or a tree. Proof. Let H be a component of G 3 . Suppose that H is not a tree and not a cycle. Let C = x1 . . . xl x1 be a shortest cycle of H and let u be the neighbor of x1 in H not in C . The graph G − ( E (C ) ∪ {x1 u }) decomposes into F ∞ and F 4 . We may assume that each 3-vertex on C − x1 has its incident edge (in G − ( E (C ) ∪ {x1 u })) in F ∞ , since otherwise we can move this incident edge from F 4 to F ∞ . Now adding x1 x2 in F ∞ and all the remaining edges in F 4 extends the decomposition to G, a contradiction. 2 We use now a discharging method to show that every graph satisfying (a), (b), and (c) of Lemma 1 and Lemma 2 70 has average degree at least 23 , and hence there is no counterexample to (T1). To apply the discharging method, we first give each vertex a “charge” equal to its degree. We then redistribute the charge (without changing the total 70 charge) to obtain charge at least 23 on each vertex. To facilitate the discharging argument, we also move some charge to special subgraphs, called banks and cores (defined below). They start with charge 0 and will end with non-negative charge, so the initial average degree is at least 70 . 23 Given such a graph G, let X be the set of all 6-vertices in G that are adjacent to exactly six vertices of degree 2, and let Y be the set of all 2-vertices adjacent to at least one vertex of X . Define the bank of type 1 of G to be the maximal bipartite subgraph of G with partite sets X and Y . A cycle in the bank of type 1 would be a (6, 2)-alternating cycle in G, which is forbidden by Lemma 1(c). Hence the bank of type 1 is a forest. We call each component of the bank of type 1 a core of type 1. By construction, each vertex of X has six neighbors in the bank; hence each leaf in the bank belongs to Y . Define the bank of type 2 of G to be the maximal subgraph of G consisted of 3-vertices. We call each component of the bank of type 2 a core of type 2. By Lemma 2, each core of type 2 is either a cycle or a tree. A vertex belonging to no core is said to be adjacent to a core C if it is adjacent in G to a vertex of C . The discharging rules are defined as follows: R1. Every 6+ -vertex gives 12 to each neighbor that is a 23 2-vertex. R2. If C is a core of type 1 or 2, v is a 4+ -vertex not belonging to a core, and v is adjacent to l vertices of C , 1 to C . then v gives l · 23 R3. Every core of type 1 gives vertices.
4 23
to each of its own 6-
M. Montassier et al. / Information Processing Letters 110 (2010) 913–916
R4. Every core of type 2 gives vertices.
1 23
to each of its own 3-
Observe that a 4+ -vertex may give charge by R1 and R2 (for example, if it is adjacent to 2-vertices and to cores). We first check that, for every vertex v, the new charge 70 ω∗ ( v ) of v after the discharging procedure is at least 23 . By Lemma 1(a), every vertex v has degree at least 2. Suppose that v is a vertex of degree 2. Vertex v starts with charge 2 and receives 12 from each adjacent 6+ 23 vertex by Lemma 1(b) and R1. Hence its new charge is ω∗ ( v ) = 2 + 2 × 12 = 70 . 23 23 Suppose that v is a vertex of degree 3. Then, by definition, it belongs to a core of type 2, say C , and so it receives 1 1 from C by R4. It follows that ω∗ ( v ) = 3 + 23 = 70 . 23 23 Suppose that v is a vertex of degree 4 or 5. Vertex v may be adjacent to cores of type 2. It follows that ω∗ ( v ) 1 d( v ) − d( v ) · 23 = d( v ) · 22 70 by R2. 23 23 Suppose that v is a vertex of degree 6. Assume first that v does not belong to a core. Then v is adjacent to at most five 2-vertices. It follows by R1 and R2, that ω∗ ( v ) 6 − 1 1 5 · 12 − 5 · 23 − 23 = 72 . Assume that v belongs to a core of 23 23 type 1, say C . Then v gives six times 4 from 23 70 . 23
C by R3. It follows that
12 23
by R1 and receives
ω∗ ( v ) = 6 − 6 ·
12 23
+
4 23
=
Suppose finally that v is a vertex of degree at least 7. 1 10 − d( v ) · 23 = d( v ) · 23 70 We have ω∗ ( v ) d( v ) − d( v ) · 12 23 23 by R1 and R2. It remains to prove that the new charge of each core is non-negative. Let C be a core of type 1. Recall that C is a tree consisted of vertices of degree 2 and 6. Let n66 (C ) and nl (C ) be the number of 66 -vertices and leaves of C respectively. Observe that:
nl (C ) 4 · n66 (C )
(1)
1 By R2, C receives nl (C ) · 23 , and gives
4 to each of its own 23 4 66 -vertices by R3 (i.e. it gives n66 (C ) · 23 ). Hence, ∗ (C ) = 1 4 1 4 nl (C ) · 23 − n66 (C ) · 23 4 · n66 (C ) · 23 − n66 (C ) · 23 = 0 by
ω
Eq. (1). Let C be a core of type 2. By Lemma 2, C is a tree or a cycle. Hence the number of edges linking C to G \ C is greater or equal to the number of vertices of C . It follows by R2 and R4 that ω∗ (C ) 0. This completes the proof of (T1). 2.2. Graphs with Mad <
192 61
The proof of (T2) is similar to the proof of (T1). We just give some parts of the proof. Assume that (T2) is false. Let G be a graph with maximum average degree less than 192 61 that does not admit an edge-partition into two forests, one having maximum degree at most 5. Moreover assume that G minimizes σ (G ) = | V (G )| + | E (G )|. By minimality of G, we have: Lemma 3. The counterexample G does not contain 1-vertices, a 2-vertex adjacent to a 6− -vertex, (7, 2)-alternating cycles; moreover each component of G 3 (the graph induced by the 3vertices) is a tree or a cycle.
915
Define now the banks and the cores. Given such a graph G, let X be the set of all 7-vertices in G that are adjacent to exactly seven vertices of degree 2, and let Y be the set of all 2-vertices adjacent to at least one vertex of X . Define the bank of type 1 of G to be the maximal bipartite subgraph of G with partite sets X and Y . A cycle in the bank of type 1 would be a (7, 2)-alternating cycle in G, which is forbidden (by Lemma 3). Hence the bank of type 1 is a forest. We call each component of the bank of type 1 a core of type 1. By construction, each vertex of X has seven neighbors in the bank; hence each leaf in the bank belongs to Y . The bank of type 2 is identical as one of the proof of (T1): define the bank of type 2 of G to be the maximal subgraph of G consisted of 3-vertices. We call each component of the bank of type 2 a core of type 2. Each core of type 2 is either a cycle or a tree. (which To show that G has average degree at least 192 61 is a contradiction), we assign to each vertex a charge equal to its degree and to each core a charge equal to 0; we then apply the following discharging rules: to each neighbor that is a R1. Every 7+ -vertex gives 35 61 2-vertex. R2. If C is a core of type 1, v is a 7+ -vertex belonging to no core, and v is adjacent to l vertices of C , then v 2 to C . gives l · 61 If C is a core of type 2, v is a 4+ -vertex belonging to no core, and v is adjacent to l vertices of C , then v 9 gives l · 61 to C . R3. Every core of type 1 gives tices. R4. Every core of type 2 gives tices.
10 61
to each of its own 7-ver-
9 61
to each of its own 3-ver-
Vertices of G have degree at least 2 by Lemma 3. A vertex v of degree 2 is adjacent to two 7+ -vertices and so by R1; it follows that ω∗ ( v ) = receives two times 35 61 2+2·
35 61
=
192 . 61
A vertex v of degree 3 belongs to a
core C of type 2 by definition, hence v receives C by R4 and ω∗ ( v )
=3+
9 61
=
192 . 61
9 61
from
A vertex v of degree k = 4, 5, 6 can give charges only by R2; it follows that 9 ω∗ ( v ) k − k · 61 208 . Now let v be a vertex of degree 7. 61 Assume first that v does not belong to a core. Then v is adjacent to at most six 2-vertices. It follows by R1 and R2, 2 9 − 6 · 61 − 61 = 196 . Assume that that ω∗ ( v ) 7 − 6 · 35 61 61 v belongs to a core of type 1, say C . Then v gives seven times 35 by R1 and receives 10 from C by R3. It follows 61 61 that
ω∗ ( v ) = 7 − 7 ·
35 61
+
10 61
=
192 . 61
Finally a vertex v of
degree k 8 may give at most k times that ω∗ ( v )
k−k·
35 61
+
2 61
=
24 61
35 61
·k
2 . It follows 61 192 . It remains 61
+
to see that each core has a non-negative final charge. Let C be a core of type 1. Recall that C is a tree consisted of vertices of degree 2 and 7. Let n77 (C ) and nl (C ) be the number of 77 -vertices and leaves of C respectively. Ob2 , serve that: nl (C ) 5 · n77 (C ). By R2, C receives nl (C ) · 61 and gives
10 61
gives n77 (C ) · 2 61
to each of its own 77 -vertices by R3 (i.e. it 10 ). 23
Hence, 10 61
2 ω∗ (C ) = nl (C ) · 61 − n77 (C ) · 10 61
5 · n 77 ( C ) · − n77 (C ) · = 0. Let C be a core of type 2; C is a tree or a cycle. Hence the number of edges linking C
916
M. Montassier et al. / Information Processing Letters 110 (2010) 913–916
to G \ C is greater or equal to the number of vertices of C . It follows by R2 and R4 that ω∗ (C ) 0. This completes the proof of (T2). This approach does not work for higher values of k. In fact, a 4-vertex may give to much charge to 3-vertices. For example, when k = 6, we have mk = 42 . A 3-vertex 13 needs
3 . A 4-vertex v may 13 3 = 40 on v which is 13 13
give 4 times
3 . 13
It remains
4−4· less than mk . A possible way to extend this result for k 6 may be to define a new bank of type 2 including vertices of degree 3 and 4. References [1] J. Balogh, M. Kochol, A. Pluhár, X. Yu, Covering planar graphs with forests, J. Combin. Theory Ser. B 94 (2005) 147–158. [2] O.V. Borodin, S.G. Hartke, A.O. Ivanova, A.V. Kostochka, D.B. West, (5, 2)-Coloring of sparse graphs, Sib. Elektron. Mat. Izv. 5 (2008) 417–426, http://semr.math.nsc.ru. [3] O.V. Borodin, A.O. Ivanova, Near-proper list vertex 2-colorings of sparse graphs, Diskretn. Anal. Issled. Oper. 16 (2) (2009) 16–20 (in Russian). [4] O.V. Borodin, A.O. Ivanova, A.V. Kostochka, Oriented vertex 5-coloring of sparse graphs, Diskretn. Anal. Issled. Oper. 13 (1) (2006) 16–32 (in Russian).
[5] O.V. Borodin, A.O. Ivanova, M. Montassier, P. Ochem, A. Raspaud, Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k, J. Graph Theory (2009), doi:10.1002/jgt.20467, in press. [6] O.V. Borodin, A.V. Kostochka, N.N. Sheikh, G. Yu, Decomposing a planar graph with girth 9 into a forest and a matching, European J. Combin. 29 (5) (2008) 1235–1241. [7] D. Gonçalves, Covering planar graphs with forests, one having bounded maximum degree, J. Combin. Theory Ser. B 99 (2009) 314– 322. [8] F. Havet, J.-S. Sereni, Improper choosability of graphs and maximum average degree, J. Graph Theory 52 (2006) 181–199. [9] W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307–317. [10] D. Kleitman, Partitioning the edges of a girth 6 planar graph into those of a forest and those of a set of disjoint paths and cycles, Manuscript, 2006. [11] M. Montassier, A. Pêcher, A. Raspaud, D. West, X. Zhu, Decomposition of sparse graphs, with application to game coloring number, Discrete Math. 310 (10–11) (2010) 1520–1523. [12] C.St.J.A. Nash-Williams, Decompositions of finite graphs into forests, J. London Math. Soc. 39 (12) (1964). [13] X. Zhu, The game coloring number of planar graphs, J. Combin. Theory Ser. B 75 (1999) 245–258.