Equitable partition of plane graphs with independent crossings into induced forests
EEquitable partition of plane graphs with independentcrossings into induced forests
Bei Niu , Xin Zhang ∗† , Yuping Gao ‡
1. School of Mathematics and Statistics, Xidian University, Xi’an 710071, China2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
December 17, 2019
Abstract
The cluster of a crossing in a graph drawing in the plane is the set of the four end-verticesof its two crossed edges. Two crossings are independent if their clusters do not intersect. In thispaper, we prove that every plane graph with independent crossings has an equitable partitioninto m induced forests for any m ≥
8. Moreover, we decrease this lower bound 8 for m to 6, 5,4 and 3 if we additionally assume that the girth of the considering graph is at least 4, 5, 6 and26, respectively. Keywords : equitable partition; vertex arboricity; planar graph; IC-planar graph
All graphs considered in this paper are finite and simple unless otherwise stated. By V ( G ), E ( G ), δ ( G ) and ∆ ( G ), we denote the vertex set, the edge set, the minimum degree and the maximumdegree of a graph G , respectively. In this paper, | G | stands for | V ( G ) | , and e ( G ) stands for | E ( G ) | .For two disjoint subsets S and S of V ( G ), E ( S , S ) (resp. e ( S , S )) is the set (resp. number) ofedges that have one end-vertex in S and another in S . Under this notation, if S consists of onlyone vertex v , then we use e ( v, S ) instead of e ( { v } , S ). The girth g ( G ) of a graph G is the length Emails: B. Niu ([email protected]), X. Zhang ([email protected]), Y. Gao ([email protected]) ∗ Corresponding author. † Supported by the National Natural Science Foundation of China (11871055) and the Youth Talent Support Planof Xi’an Association for Science and Technology (2018-6). ‡ Supported by the National Natural Science Foundation of China (11901263). a r X i v : . [ m a t h . C O ] D ec f the shortest cycle in G , and is + ∞ if G is a forest. For other undefined notation, we refer thereaders to [2].An equitable partition of a graph G is a partition of V ( G ) such that the sizes of any two partsdi ff er by at most one. In 1970, Hajnal and Szemerédi [9] answered a question of Erd ˝os by provingthat every graph G with maximum degree ∆ has an equitable partition into m independent sets forany integer m ≥ ∆ + ∆ has an equitable partition into m stable sets for any m ≥ (cid:100) ∆ (cid:101) +
1, but it admits no equitable partition into m independent sets for any m < (cid:100) ∆ (cid:101) . Therefore,finding a constant c such that every planar graph has an equitable partition into m independentsets for any m ≥ c is impossible. Surprisingly, if we ask for an equitable partition into inducedforests rather than stable sets, we succeed. In 2005, Esperet, Lemoine and Ma ff ray [7] confirmeda conjecture of Wu, Zhang and Li [14] by proving the following theorem. Theorem 1.1.
Every planar graph has an equitable partition into m induced forests for any m ≥ if it can be drawn in the plane so that each edge is crossed by at mostone other edge, and a drawing satisfying this property so that the number of crossings is as fewas possible is a . The notion of 1-planarity was introduced by Ringel [13] whiletrying to simultaneously color the vertices and faces of a plane graph G such that any pair of ad-jacent / incident elements receive di ff erent colors. Ringel [13] showed that every 1-planar graph is7-colorable, and Borodin [3, 4] improved it to the 6-colorability. Recently in 2017, Kobourov, Li-otta and Montecchiani [10] reviewed the current literature covering various research streams about1-planarity, such as characterization and recognition, combinatorial properties, and geometric rep-resentations.Clearly, every crossing c in a 1-plane graph G is generated by two mutually crossed edges e and e . Thus, for every crossing c there exists a vertex set M G ( c ) of size four, where M G ( c ), the cluster of c , consists of the end-vertices of e and e . For two distinct crossings c and c in a1-plane graph G , it is clear that | M G ( c ) ∩ M G ( c ) | ≤ G be a 1-plane graph. If M G ( c ) ∩ M G ( c ) = ∅ for any two distinct crossings c and c , then G is a plane graph with independent crossings ( IC-plane graph , for short). A graph that admitsa drawing homeomorphic to an IC-plane graph is an
IC-planar graph . The IC-planarity was firstconsidered by Albertson [1] in 2008, who conjectured that every IC-planar graph is 5-colorable.This conjecture was confirmed by Král and Stacho [11] in 2010. Note that IC-planar graph can benon-planar. 2n this paper, we consider the equitable partition problem of IC-planar graphs by proving thefollowing.
Theorem 1.2.
Every plane graph with independent crossings and with girth at least g has an equi-table partition into m induced forests for any m ≥ F ( g ), where F ( g ) = , if g = , if g = , if g = , if g = , if g = If a graph G has an equitable partition into m induced forests, we say that G is equitably tree-m-colorable , and has an equitable tree-m-coloring . Let G g be the class of IC-plane graph with girthat least g . Note that G ⊇ G ⊇ G ⊇ . . . ⊇ G + ∞ . Lemma 2.1. If G ∈ G g , then e ( G ) ≤ g − g − | G | − gg − . Proof.
Since every IC-plane graph G has at most (cid:98) | G |(cid:99) crossings by its definition, we can obtain aplane graph G (cid:48) with order | G | via removing at most (cid:98) | G |(cid:99) edges from G . Since g ( G (cid:48) ) ≥ g ( G ) ≥ g , e ( G (cid:48) ) ≤ gg − (cid:0) | G | − (cid:1) by the famous Euler’s formula. Therefore, the required result holds since e ( G ) ≤ e ( G (cid:48) ) + | G | . (cid:3) Lemma 2.2.
Let G be a graph in G g .(a) If g =
3, then δ ( G ) ≤ g =
4, then δ ( G ) ≤ g ≥
5, then δ ( G ) ≤ Proof.
The average degree d ( G ) of G is 2 e ( G ) / | G | , and thus is at most g − g − by Lemma 2.1. If g =
3, then d ( G ) ≤ .
5. If g =
4, then d ( G ) ≤ .
5. If g ≥
5, then d ( G ) <
4. Since δ ( G ) ≤ (cid:98) d ( G ) (cid:99) ,the results hold immediately. (cid:3) Lemma 2.3.
Let m ≥ G ( g ) be a fixed integer, where G ( g ) = (cid:40) , if g = , if g ≥ G g of order mt is equitably tree- m -colorable for any integer t ≥
1, then everygraph in G g is equitably tree- m -colorable. 3 roof. Let G be a graph in G g with order n . If n ≤ m , then it is trivial that G is equitably tree- m -colorable. Hence we assume that n > m , and next prove this lemma by induction on n (assumingthat the result holds for graphs in G g with order less than n ).If n is divisible by m , then the required result holds directly. Hence we assume that mt < n < m ( t +
1) and t ≥ v ∈ V ( G ) be a vertex with minimum degree. By the induction hypothesis, G − v has anequitable tree- m -coloring φ . Let V , V , · · · , V m be the color classes of φ , where | V i | = t or t + i ≥ n = m ( t + −
1, then we add an isolated vertex v to G . Clearly, the resulting graph G (cid:48) isan IC-plane graph of order m ( t + G (cid:48) has an equitable tree- m -coloring such that all color classes have the same size. Removing v from G (cid:48) , we obtain the graph G with an equitable tree- m -coloring.Hence in the following, we assume that n ≤ m ( t + −
2. Since | G − v | = n − ≤ m ( t + − V , V , · · · , V m there are at most m − t + g =
3, then d ( v ) ≤ m − V , V , · · · , V m satisfying | N ( v ) (cid:84) V i | ≤
1. Without loss of generality, assume that | N ( v ) (cid:84) V i | ≤ ≤ i ≤ m . If | V i | = t for some i ≥
4, then by adding v to V i , we get an equitable tree- m -coloring of G (with color classes V , · · · , V i − , V i (cid:83) { v } , V i + , · · · , V m ). Hence we assume that | V i | = t + i ≥
4. This implies that | V | = | V | = | V | = t , since | G − v | ≤ m ( t + − u ∈ (cid:83) mi = V i such that e ( u , V j ) ≤ ≤ j ≤
3, then by transferring v to the color class containing u , and adding u to V j , we get an equitable tree- m -coloring of G .Hence, for any u ∈ (cid:83) mi = V i and any V j with 1 ≤ j ≤
3, we have e ( u , V j ) ≥
2. This implies e ( G ) ≥ t + m − G [ E ( V (cid:83) V (cid:83) V , (cid:83) mi = V i )] is a bipartite IC-plane graph (so it hasgirth at least 4), e ( G ) ≤ [ m ( t + − − [ m ( t + − − ≥ t + m − m ≥ g ≥
4, then d ( v ) ≤ m − V , V , · · · , V m satisfying | N ( v ) (cid:84) V i | ≤
1. Without loss of generality, assume that | N ( v ) (cid:84) V i | ≤ ≤ i ≤ m . Since | G − v | ≤ m ( t + −
3, among V , · · · , V m , there is at least one class, say V ,containing exactly t vertices. Therefore, by moving v to V , we obtain an equitable tree- m -coloringof G . (cid:3) The structures of the edge-minimal counterexample
Let G be an edge-minimal graph with | G | = mt in the class G g that is not equitably tree- m -colorable.Here we assume that m ≥ g = m ≥ g = m ≥ g = m ≥ g =
6, and m ≥ g ≥
7. This section is devoted to exploring the structures of G , which will be later used to proveTheorem 1.2 by contradiction in the next section.Clearly, G contains a vertex of degree at least 1. Let δ ( g ) = , if g = , if g = , if g ≥ δ ( G ) ≤ δ ( g ) by Lemma 2.2, there is an edge xx ∈ E ( G ) with 1 ≤ d ( x ) ≤ δ ( g ). By theminimality of G , G − xx admits an equitable tree- m -coloring with m color classes V , V , . . . , V m ,each of which has size t .Clearly, xx is contained in a cycle of the subgraph induced by some color class, for otherwisethe current coloring of G − xx is just an equitable tree- m -coloring of G . Therefore, x , x andanother neighbor of x , say x , is contained in a same color class, say V , and then we assume that N ( x ) ⊆ ∪ δ ( g ) − i = V i . Let V (cid:48) = V \ { x } .If g =
3, then d ( x ) ≤
6. Since x has two neighbors contained in V , among V , V , V and V , at most two of them contains at least two neighbors of x . Hence we assume, without loss ofgenerality, that | N ( x ) ∩ V | ≤ | N ( x ) ∩ V | ≤ g =
4, then d ( x ) ≤
4. Since x has two neighbors contained in V , among V and V , at mostone of them contains at least two neighbors of x . Hence we assume, without loss of generality, that | N ( x ) ∩ V | ≤ g ≥
5, then d ( x ) ≤
3. Since x has two neighbors contained in V , | N ( x ) ∩ V | ≤ Claim 1. (a) If G ∈ G , then e ( v, V (cid:48) ) ≥ v ∈ ∪ mi = V i .(b) If G ∈ G , then e ( v, V (cid:48) ) ≥ v ∈ ∪ mi = V i .(c) If G ∈ G g with g ≥
5, then e ( v, V (cid:48) ) ≥ v ∈ ∪ mi = V i . Proof.
We just prove (a), and another two results can be similarly verified. Suppose, to the con-trary, that there exists v ∈ V i for some i ≥ e ( v, V (cid:48) ) ≤ . By transferring v from V i to V (cid:48) and adding x to V i \ { v } , we get an equitable tree- m -coloring of G , a contradiction. (cid:3) Claim 2. (a) If G ∈ G and m ≥
5, then for every v ∈ V ∪ V , e ( v, V (cid:48) ) ≥ G ∈ G and m ≥
5, then for every v ∈ V , e ( v, V (cid:48) ) ≥ Proof.
We just prove (a). Note that (b) is a corollary of (a) since G ⊆ G .5uppose, to the contrary, that there exists w ∈ V such that e ( w, V (cid:48) ) ≤
1. In this case, e ( v, V ) ≥ v ∈ m (cid:91) i = V i . (3.1)Otherwise, suppose that e ( v, V ) ≤ v ∈ V i with 4 ≤ i ≤ m . Transferring v from V i to V , w from V to V (cid:48) and adding x to V i \ { v } , we get an equitable tree- m -coloring of G , a contradiction.If there exists w (cid:48) ∈ V such that e ( w (cid:48) , V ) ≤
1, then e ( v, V ) ≥ v ∈ (cid:83) mi = V i . Otherwise,suppose that e ( v, V ) ≤ v ∈ V i with 4 ≤ i ≤ m . Transferring v from V i to V , w (cid:48) from V to V , w from V to V (cid:48) and adding x to V i \ { v } , we get an equitable tree- m -coloring of G , acontradiction.If there exists w (cid:48) ∈ V such that e ( w (cid:48) , V (cid:48) ) ≤
1, then e ( v, V ) ≥ v ∈ (cid:83) mi = V i . Otherwise,suppose that e ( v, V ) ≤ v ∈ V i with 4 ≤ i ≤ m . Transferring v from V i to V , w (cid:48) from V to V (cid:48) and adding x to V i \ { v } , we get an equitable tree- m -coloring of G , a contradiction.In each of the above two cases, by Claim 1 and by (3.1), we have e ( (cid:83) mi = V i , V (cid:48) (cid:83) V (cid:83) V ) ≥ m − t . Since G [ E ( (cid:83) mi = V i , V (cid:48) (cid:83) V (cid:83) V )] is a bipartite IC-plane graph of order mt −
1, we have e ( (cid:83) mi = V i , V (cid:48) (cid:83) V (cid:83) V ) ≤ ( mt − − = mt − by Lemma 2.1. Since m ≥
5, 6( m − t > mt − ,a contradiction, too. Hence e ( w (cid:48) , V ) ≥ e ( w (cid:48) , V (cid:48) ) ≥ w (cid:48) ∈ V . (3.2)By Claim 1, (3.1), and (3.2), we conclude that e ( (cid:83) mi = V i , V (cid:48) (cid:83) V ) ≥ m − t .Since G [ E ( (cid:83) mi = V i , V (cid:48) (cid:83) V )] is a bipartite IC-plane graph of order mt − e ( (cid:83) mi = V i , V (cid:48) (cid:83) V ) ≤ ( mt − − = mt − by Lemma 2.1. Since m ≥
5, 4( m − t > mt − , a contradiction. Hence, e ( w, V (cid:48) ) ≥ w ∈ V . By similar argument as above, we conclude that e ( w (cid:48) , V (cid:48) ) ≥ w (cid:48) ∈ V . (cid:3) Let A = ∪ mi = V i . By Claims 1 and 2, if G ∈ G and m ≥
5, or G ∈ G , then e ( v, V (cid:48) ) ≥ v ∈ A , and thus e ( A , V (cid:48) ) ≥ m − t . (3.3)Therefore, we divide A into two parts, say A and A \ A , where A = { v ∈ A | e ( v, V (cid:48) ) = } . Let r = | A | , then e ( A , V (cid:48) ) ≥ r + (cid:0) ( m − t − r (cid:1) = m − t − r . (3.4)Next, we calculate the lower bound for r . Since G [ E ( A , V (cid:48) )] is a bipartite IC-plane graph (soodd cycles are forbidden) and is also a subgraph of G , its girth g is an even integer no less than g .Hence g ≥ g ≤
4, and g ≥ g ≥
5. 6y (3.4) and Lemma 2.1,5 g − g − mt − − g g − ≥ e ( A , V (cid:48) ) ≥ m − t − r , which implies that r ≥ (cid:0) m − (cid:1) t + , if g = g = (cid:0) m − (cid:1) t + , if g ≥
5. (3.5)
Lemma 3.1.
There exists a vertex z ∈ V (cid:48) that has two nonadjacent neighbors y , y in A if one ofthe following conditions is satisfied:(i) r > t −
1) and g = r > ( t −
1) and g ≥ e ( A , V (cid:48) ) ≤ (3 m − t + g = G ∈ G and m ≥ G ∈ G and m ≥ G ∈ G and m ≥ Proof. (i) Suppose that for each vertex z ∈ V (cid:48) , e ( z , A ) ≤
4. Since | V (cid:48) | = t − r > t − t − ≥ e ( A , V (cid:48) ) = r > t − z ∈ V (cid:48) , such that e ( z , A ) ≥
5. Since K is not an IC-plane graph, there are two neighbors of z in A that are notadjacent, and thus the required structure occurs.(ii) Suppose that for each vertex z ∈ V (cid:48) , e ( z , A ) ≤
1. Since | V (cid:48) | = t − r > ( t − t − ≥ e ( A , V (cid:48) ) = r > ( t − z ∈ V (cid:48) , suchthat e ( z , A ) ≥
2. Since K is forbidden in an IC-plane graph with girth at least 4, there are twoneighbors of z in A that are not adjacent, and thus the required structure occurs.(iii) In this case, by (3.4), (3 m − t + ≥ e ( A , V (cid:48) ) ≥ m − t − r , which implies r > t − m ≥
7, then by (3.5), r ≥ ( × − t + > t −
1) and (i) is satisfied.(v) If m ≥
5, then by (3.5), r ≥ ( × − t + > ( t −
1) and (ii) is satisfied.(vi) If m ≥
3, then by (3.5), r ≥ ( × − t + > ( t −
1) and (ii) is satisfied. (cid:3)
Suppose that there exists a vertex z ∈ V (cid:48) that has two nonadjacent neighbors y , y in A . It iseasy to see that V (cid:48) ∪ { y , y } \ { z } induces a forest F of order t . Let G (cid:48) be the graph induced by A ∪ { x , z } \ { y , y } . Note that | G (cid:48) | = | A | − + = ( m − t .7 laim 3. e ( G (cid:48) ) ≤ e ( G ) − ( m − t − Proof.
Since e ( v, V (cid:48) ) ≥ v ∈ A , we have e ( v, V (cid:48) \ { z } ) ≥ v ∈ A \ { y , y } and e ( A \ { y , y } , V (cid:48) \ { z } ) ≥ | A \ { y , y }| = ( m − t −
2. Counting the four edges xx , xx , z y , z y ,we immediately have e ( G (cid:48) , F ) ≥ ( m − t − + = ( m − t +
2. This implies that e ( G (cid:48) ) ≤ e ( G ) − ( m − t − (cid:3) Claim 4. If G (cid:48) is equitably tree-( m − G is equitably tree- m -colorable. Proof.
Since | G (cid:48) | = ( m − t , G (cid:48) has an equitable partition into m − F , . . . , F m with | F i | = t for each 2 ≤ i ≤ m . It follows that G has an equitable partition into m inducedforests F , F , . . . , F m , a contradiction to the choice of G . Recall that F is the graph induced by V (cid:48) (cid:83) { y , y }\{ z } , which is a forest of order t . (cid:3) In the proofs of the following theorems, we use the edge-minimal-counterexample-arguments asmentioned in Section 3, and thus the notations and results in Section 3 can be applied here.
Theorem 4.1.
Let s ∈ { , , , } . If G is a graph in G of order mt and size at most (cid:18) s − m + − s + s − (cid:19) t − (22 − s ) , then G has an equitable partition into m induced forests for any m ≥ s . Proof.
We prove it by induction on s . First of all, if s =
5, then e ( G ) ≤ ( m + ) t −
12, whichimplies by (3.3) that e ( A , V (cid:48) ) − e ( G ) ≥ m − t − (cid:0) ( m + ) t − (cid:1) > m ≥
5, a contradiction.We assume that the result holds for s = k −
1, where 6 ≤ k ≤
8. Now we consider the casewhen s = k .If s ∈ { , } , then by Lemma 3.1(iv), there exists a vertex z ∈ V (cid:48) that has two nonadjacentneighbors y , y in A . If s =
6, then e ( A , V (cid:48) ) ≤ e ( G ) ≤ ( m + ) t − ≤ (3 m − t +
1, and thusby Lemma 3 . e ( G (cid:48) ) ≤ e ( G ) − ( m − t − ≤ (cid:18) k − m + − k + k − (cid:19) t − (22 − k ) − ( m − t − = (cid:18) k − −
194 ( m − + − k − + k − − (cid:19) t − (cid:0) − k − (cid:1) . G (cid:48) is an IC-plane graph and m − ≥ s − = k − G (cid:48) admits an equitable tree-( m − G admits an equitable tree- m -coloring. (cid:3) Theorem 4.2.
Let s ∈ { , , } . If G is a graph in G of order mt and size at most (cid:18) s − m + − s + s − (cid:19) t − (16 − s ) , then G has an equitable partition into m induced forests for any m ≥ s . Proof.
We prove it by induction on s . First of all, if s =
4, then e ( G ) ≤ ( m + ) t −
8, whichimplies by (3.3) that e ( A , V (cid:48) ) − e ( G ) ≥ m − t − (cid:0) ( m + ) t − (cid:1) > m ≥
4, a contradiction.We assume that the result holds for s = k −
1, where 5 ≤ k ≤
6. Now we consider the casewhen s = k .If s ∈ { , } , then by Lemma 3 . z ∈ V (cid:48) that has two nonadjacentneighbors y , y in A . Therefore, by Claim 3, we have e ( G (cid:48) ) ≤ e ( G ) − ( m − t − ≤ (cid:18) k − m + − k + k − (cid:19) t − (16 − k ) − ( m − t − = (cid:18) k − −
154 ( m − + − k − + k − − (cid:19) t − (cid:0) − k − (cid:1) . Since G (cid:48) is an IC-plane graph and m − ≥ s − = k − G (cid:48) admits an equitable tree-( m − G admits an equitable tree- m -coloring. (cid:3) Choosing s to be 8 and 6 in Theorem 4.1 and in Theorem 4.2, respectively, we conclude byLemmas 2.1 and 2.3 that Theorem 4.3.
Every plane graph with independent crossings has an equitable partition into m induced forests for each m ≥ (cid:3) Theorem 4.4.
Every plane graph with independent crossings and with girth at least 4 has an equi-table partition into m induced forests for each m ≥ (cid:3) Now, we consider plane graph with independent crossings and with higher girth.
Theorem 4.5.
Every plane graph with independent crossings and with girth at least 5 has an equi-table partition into m induced forests for each m ≥ roof. By Lemma 2.3, we assume that the order of the considering graph G is divided by m , thatis, | G | = mt . By Lemma 2.1, we have e ( G ) ≤ mt − . Since m ≥
5, there exists a vertex z ∈ V (cid:48) that has two nonadjacent neighbors y , y in A byLemma 3 . e ( G (cid:48) ) ≤ e ( G ) − ( m − t − ≤ mt − − ( m − t − = (cid:18) m + (cid:19) t − . (4.1)Now, by Claim 4, proving that G (cid:48) admits an equitable tree-( m − G (cid:48) , we immediately have, by (3.3), that e ( G (cid:48) ) ≥ (cid:0) ( m − − (cid:1) t = (2 m − t . Hence by (4.1), we have (cid:18) m + (cid:19) t − ≥ (2 m − t , which implies that m ≤
4, a contradiction. (cid:3)
Theorem 4.6.
Every plane graph with independent crossings and with girth at least 6 has an equi-table partition into m induced forests for each m ≥ Proof.
By Lemma 2.3, we assume that the order of the considering graph G is divided by m , thatis, | G | = mt . By Lemma 2.1, we have e ( G ) ≤ mt − . Since m ≥
4, there exists a vertex z ∈ V (cid:48) that has two nonadjacent neighbors y , y in A byLemma 3 . e ( G (cid:48) ) ≤ e ( G ) − ( m − t − ≤ mt − − ( m − t − = (cid:18) m + (cid:19) t − . (4.2)Now, by Claim 4, proving that G (cid:48) admits an equitable tree-( m − G (cid:48) , we immediately have, by (3.3), that e ( G (cid:48) ) ≥ (cid:0) ( m − − (cid:1) t = (2 m − t . (cid:18) m + (cid:19) t − ≥ (2 m − t , which implies that m ≤
3, a contradiction. (cid:3)
Theorem 4.7.
Every plane graph with independent crossings and with girth at least 26 has anequitable partition into m induced forests for each m ≥ Proof.
By Lemma 2.3, we assume that the order of the considering graph G is divided by m , thatis, | G | = mt . By Lemma 2.1, we have e ( G ) ≤ mt − . Hence by (3.3) we conclude that e ( A , V (cid:48) ) − e ( G ) ≥ m − t − ( mt − ) > m ≥ G admits anequitable tree- m -coloring. (cid:3) The proof of Theorem 1.2.
See Theorems 4.3, 4.4, 4.5, 4.6 and 4.7, respectively. (cid:3)
Formerly, the minimum integer k such that G has an equitable partition into k induced forests isthe equitable vertex arboricity of G , denoted by v a eq ( G ), and the minimum integer k such that G has an equitable partition into m induced forests for any m ≥ k is the equitable vertex arborablethreshold of G , denoted by v a ∗ eq ( G ). Theorem 1.2 actually implies that v a ∗ eq ( G ) ≤ F ( g ) if G is aplane graph with independent crossings and with girth at least g . Precisely, choosing g =
3, weconclude that v a ∗ eq ( G ) ≤ G is a plane graph with independent crossings. Here, we do not knowwhether the upper bound 8 for v a ∗ eq ( G ) is sharp (actually we think that it may be improved), butthis bound is acceptable at this stage, since 8 is a constant not very large. Note that the paper ofEsperet, Lemoine and Ma ff ray [7] implies that v a ∗ eq ( G ) ≤
19 if G is a 1-planar graph, whose acyclicchromatic number is at most 20 [5].In 2013, Wu, Zhang and Li [14] put forward two conjectures in their paper. Although Esperet,Lemoine and Ma ff ray [7] solved one in 2015, the other (Conjecture 5.1) is still open. Conjecture 5.1. v a ∗ eq ( G ) ≤ (cid:100) ∆ ( G ) + (cid:101) for any simple graph G .As far as we know, Conjecture 5.1 has been verified for complete graphs [14], balanced com-plete bipartite graphs [14], graphs with maximum degree ∆ ≥ ( | G | − / ∆ ≤ ∆ ≤
5) [6],and d -degenerate graphs with maximum degree ∆ ≥ d [19].Looking back to Theorem 1.2, we immediately find that Conjecture 5.1 holds for any planegraph with independent crossings and with maximum degree at least 14. Of course, we may do notlike the lower bound 14 for the maximum degree there. If we can pull this bound down to 6, thenConjecture 5.1 holds for all plane graphs with independent crossings.Note that every plane graph with independent crossings is 6-degenerate (a graph is k-degenerate if δ ( H ) ≤ k for any H ⊆ G ). Therefore, an alternate task is to prove Conjecture 5.1 for all 6-degenerate graphs directly. Actually, we propose the following conjecture (also see [12]). Conjecture 5.2. v a ∗ eq ( G ) ≤ k for any k -degenerate graph G .If Conjecture 5.2 can be verified, then the bound k for v a ∗ eq ( G ) is sharp. This fact can be seenfrom the graph G obtained from K k via adding t ≥ k − K k . Clearly, G is k -degenerate.If v a ∗ eq ( G ) ≤ k −
1, then G has an equitable tree-( k − ϕ . Under this coloring, twovertices of K k shall receive the same color, say 1, and all but these two vertices are not coloredwith 1, because otherwise a monochromatic triangle appears. Since ϕ is equitable, each of thecolors in { , , . . . , k − } appears at most three times in G . This implies that there are at most2 + k − = k − | G | = k + t ≥ k − d be a positive integer. An equitable d-defective tree-k-coloring of a graph G is an equitabletree- k -coloring of G such that the subgraph induced by each color class has maximum degree atmost d .The minimum integer k such that G has an equitable d -defective tree- k -coloring is the equitablevertex d-arboricity of G , denoted by v a deq ( G ), and the minimum integer k such that G has an equi-table d -defective tree- m -coloring for any m ≥ k is the equitable vertex d-arborable threshold of G ,denoted by v a ∗ deq ( G ). In 2011, Fan et al. [8] prove that v a ∗ eq ( G ) ≤ ∆ ( G ) for any graph G . Recently,Zhang and Niu [18] proved that v a ∗ eq ( G ) ≤ (cid:100) ∆ ( G ) + (cid:101) if G is a graph with ∆ ( G ) ≥ ( | G | − / ff ray mentioned (pointed out by Yair Caro, actually)that there does not exist a constant c so that v a ∗ eq ( G ) ≤ c for any planar graph G . The outer-planar graph obtained from a large path by adding a universal vertex is an example supporting thisconclusion.In fact, one can easily show for any fixed integer d ≥ v a deq ( G ) = v a ∗ deq ( G ) = (cid:100) ∆+ d (cid:101) if G isa star with maximum degree ∆ . Hence, for any fixed integer d ≥
1, finding a constant m such thatevery planar graph (even for outer-planar graph) has an equitable partition into m induced forestswith maximum degree at most d is impossible. From this point of view, the “constant” results12n the equitable vertex arboricity or the equitable vertex arborable threshold ( d = + ∞ ) of planargraphs and its relative classes are very interesting. References [1] M. O. Albertson. Chromatic number, independent ratio, and crossing number.
Ars Math.Contemp
Graph Theory . Springer, GTM 244, 2008.[3] O. V. Borodin. Solution of Ringel’s problems on the vertex-face coloring of plane graphs andon the coloring of 1-planar graphs.
Diskret. Analiz
41 (1984) 12–26.[4] O. V. Borodin. A new proof of the 6-color theorem.
J. Graph Theory
Discrete Appl. Math.
114 (2001) 29–41.[6] G. Chen, Y. Gao, S. Shan, G. Wang, J.-L. Wu. Equitable vertex arboricity of 5-degenerategraphs.
J. Comb. Optim. ff ray. Equitable partition of graphs into induced forests. Dis-crete Math.
338 (2015) 1481–1483.[8] H. Fan, H. A. Kierstead, G. Liu, T. Molla, J.-L. Wu, X. Zhang. A note on relaxed equitablecoloring of graph.
Inform. Process. Lett.
111 (2011) 1062–1066.[9] A. Hajnal, E. Szemerédi. Proof of a conjecture of P. Erd ˝os. In:
Combinatorial Theory andits Applications (P. Erd ˝os, A. Rényi and V. T. Sós, eds), North-Holand, London. 1970 pp.601–623.[10] S. G. Kobourov, G. Liotta, F. Montecchiani. An annotated bibliography on 1-planarity.
Com-put. Sci. Rev.
25 (2017) 49–67.[11] D. Král, L. Stacho. Coloring plane graphs with independent crossings.
J. Graph Theory
J. Comb. Optim. , doi:10.1007 / s10878-019-00461-7.[13] G. Ringel. Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Semin. Univ. Hambg.
Discrete Math
313 (23)(2013) 2696–2701.[15] X. Zhang. Drawing complete multipartite graphs on the plane with restrictions on crossings.
Acta Math. Sin. (Engl. Ser.)
Taiwanese J. Math
19 (1) (2015)123–131.[17] X. Zhang. Equitable vertex arboricity of subcubic graphs.
Discrete Math.
339 (2016) 1724–1726.[18] X. Zhang, B. Niu, Equitable partition of graphs into induced linear forests.