On the sizes of bipartite 1-planar graphs

A graph is called $1$-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let $G$ be a bipartite 1-planar graph with $n$ ($\ge 4$) vertices and $m$ edges. Karpov showed that $m\le 3n-8$ holds for even $n\ge 8$ and $m\le 3n-9$ holds for odd $n\ge 7$. Czap, Przybylo and \u{S}krabul\'{a}kov\'{a} proved that if the partite sets of $G$ are of sizes $x$ and $y$, then $m\le 2n+6x-12$ holds for $2\leq x\leq y$, and conjectured that $m\le 2n+4x-12$ holds for $x\ge 3$ and $y\ge 6x-12$. In this paper, we settle their conjecture and our result is even under a weaker condition $2\le x\le y$.


Introduction
A drawing of a graph G = (V, E) is a mapping D that assigns to each vertex in V a distinct point in the plane and to each edge uv in E a continuous arc connecting D(u) and D(v). We often make no distinction between a graph-theoretical object (such as a vertex, or an edge) and its drawing. All drawings considered here are such ones that no edge crosses itself, no two edges cross more than once, and no two edges incident with the same vertex cross. The crossing number of a graph G is the smallest number of crossings in any drawing of G.
A drawing of a graph is 1-planar if each of its edges is crossed at most once. If a graph has a 1-planar drawing, then it is 1-planar. The notion of 1-planarity was introduced in 1965 by Ringel [11], and since then many properties of 1-planar graphs have been studied (e.g. see the survey paper [8]).
It is well-known that any simple planar graph with n (n ≥ 3) vertices has at most 3n − 6 edges, and a simple and bipartite graph with n (n ≥ 3) vertices has at most 2n − 4 edges. Determining the maximum number of edges in 1-planar graphs with a fixed number of vertices has aroused great interest of many authors (see, for example, [2], [4], [6], [10], [13]). It is known that [2, 6, 10] any 1-planar graph with n (≥ 3) vertices has at most 4n − 8 edges. For bipartite 1-planar graphs, an analogous result was due to Karpov [7]. Theorem 1 ( [7]) Let G be a bipartite 1-planar graph with n vertices. Then G has at most 3n − 8 edges for even n = 6, and at most 3n − 9 edges for odd n and for n = 6. For all n ≥ 4, these bounds are tight.
Note that Karpov's upper bound on the size of a bipartite 1-planar graph is in terms of its vertex number. When the sizes of partite sets in a bipartite 1-planar graph are taken into account, Czap, Przybylo andSkrabuláková [5] obtained another upper bound for its size (i.e., Corollary 2 in [5]).
For any graph G, let V (G) and E(G) denote its vertex set and edge set.
For each pair of integers x and y with x ≥ 3 and y ≥ 6x − 12, the authors in [5] constructed a bipartite 1-planar graph G with partite sets of sizes x and y such that |E(G)| = 2|V (G)| + 4x − 12 holds. Moreover, they believed this lower bound is optimal for such graphs and thus posed the following conjecture. Conjecture 1 ( [5]) For any integers x and y with x ≥ 3 and y ≥ 6x − 12, if G is a bipartite 1-planar graph with partite sets of sizes x and y, then |E(G)| ≤ 2|V (G)| + 4x − 12.
In this paper we obtain the following result which proves Conjecture 1.
Theorem 3 Let G be a bipartite 1-planar graph with partite sets of sizes x and y, where 2 ≤ x ≤ y. Then |E(G)| ≤ 2|V (G)| + 4x − 12, and the upper bound is best possible.
The result in [5,Lemma 4] shows that the upper bound for |E(G)| in Theorem 3 is tight. Also, if x ≤ 1 3 (y + 4), Theorem 3 provides a better upper bound for |E(G)| than Theorem 1.
The authors in [5] mentioned a question of Sopena [12]: How many edges we have to remove from the complete bipartite graph with given sizes of the partite sets to obtain a 1-planar graph? It is not hard to see that Theorem 3 implies the follow corollary which answers the problem.
Corollary 1 Let K x,y be the complete bipartite graph with partite sets of sizes x and y, where 2 ≤ x ≤ y. Then at least (x − 2)(y − 6) edges must be removed from K x,y such that the resulting graph becomes possibly a 1-planar graph, and the lower bound on the number of removed edges is best possible.
The remainder of this paper is arranged as follows. In Section 2, we explain some terminology and notation used in this paper. In Section 3, under some restrictions, we present several structural properties on an extension of D × for a 1-planar drawing D of a bipartite 1-planar graph G, where D × is a plane graph introduced in Section 2. Some important lemmas for proving Theorem 3 are given in Section 4, while the proof of this theorem is completed in Section 5. Finally, we give some further problems in Section 6.

Terminology and notation
All graphs considered here are simple, finite and undirected, unless otherwise stated. For terminology and notation not defined here, we refer to [1]. For any graph G and A ⊆ V (G), let G[A] denote the subgraph of G with vertex set A and edge set {e ∈ E(G) : e joins two vertices in A}.
A walk in a graph is alternately a vertex-edge sequence; the walk is closed if its original vertex and terminal vertex are the same. A path (respectively, a cycle ) of a graph is a walk (respectively, a closed walk) in which all vertices are distinct; the length of a path or cycle is the number of edges contained in it. A path (respectively, a cycle) of length k is said to be a k-path (respectively, k-cycle). If a cycle C is composed of two paths P 1 and P 2 , we sometimes write C = P 1 ∪ P 2 .
A plane graph is a planar graph together with a drawing without crossings, and at this time we say that G is embedded in the plane. A plane graph G partitions the plane into a number of connected regions, each of which is called a face of G. If a face is homeomorphic to an open disc, then it is called cellular; otherwise, noncellular. Actually, a noncellular face is homeomorphic to an open disc with a few removed "holes". For a cellular face f , the boundary of f can be regarded as a closed walk of G, while for a noncellular face f , its boundary consists of many disjoint closed walks of G. The size of a face is the number of the edges contained in its the boundary with each repeated edge counts twice. A face of size k is also said to be a k-face.
It is known that a plane graph G has no noncellular faces if and only if G is connected. For a connected plane graph G, the well-known Euler's formula states that |V (G)| − |E(G)| + |F (G)| = 2, where F (G) denotes the face set of G.
A cycle C of a plane graph G partitions the plane into two open regions, the bounded one (i.e., the interior of C) and the unbounded one (i.e., the exterior of C). We denote by int(C) and ext(C) the interior and exterior of C, respectively, and their closures by INT (C) and EXT (C). Clearly, INT (C) ∩ EXT (C) = C. A cycle C of a plane graph G is said to be separating if both int(C) and ext(C) contain at least one vertex of G.
Let D be a 1-planar drawing of a graph G. The associated plane graph D × is the plane graph that is obtained from D by turning all crossings of D into new vertices of degree four; these new vertices of degree four are called the crossing vertices of D × .

An extension of D ×
Throughout this section, we always assume that the considered graph G (possibly disconnected) is a bipartite 1-planar graph with partite sets X and Y , where 3 ≤ |X| ≤ |Y |. Let D be a 1-planar drawing of G with the minimum number of crossings, and D × be the associated graph of D with the crossing vertex set W .
Note that subsets X, Y and W form a partition of V (D × ). We color the vertices in X, Y and W by the black color, white color and red color respectively. As stated in [5], D × can be extended to a plane graph, denoted by D × W , by adding edges joining black vertices as described below: for each vertex w in W , it is adjacent to two black vertices in D × , say x 1 and x 2 , and we draw an edge, denoted by e w , joining x 1 and x 2 which is "most near" one side of the path x 1 wx 2 of D × such that it does not cross with any other edge, as shown in Figure 1  Observe that D × W is a plane graph with D × as its spanning subgraph and the edge set of D × W is the union of E(D × ) and {e w : w ∈ W }. Although D × is a simple graph, D × W might contain parallel edges (i.e., edges with the same pair of ends), as there may exist two edges in {e w : w ∈ W } with the same pair of ends. An example is shown in Figure 2 All vertices in H are black and the edges in H are also called black edges. Clearly, W is an independent set in F and each vertex in W (i.e., a red vertex) is of degree 2 in F . The edges in F incident with red vertices are called red edges. Thus, each edge in F is either black or red, as shown in Figure 2 (d).
We have the following facts on D × , F and H: (1) D × , F and H are simultaneously embedded in the plane; (2) F and H are obviously loopless, but they are possibly disconnected; (3) w → e w is a bijection from W to E(H), where w is a red vertex, and thus the number of crossings of D equals to |E(H)|; and (4) e w → x 1 wx 2 is a bijection from E(H) to the set of 2-paths in F whose ends are black, where w is a red vertex and x 1 and x 2 are the black vertices in D × adjacent to w.
Moreover we have the following propositions.
Proposition 1 Let e w be an edge of H with ends x 1 and x 2 and C be the 3-cycle of F consisting of e w and its corresponding 2-path P = x 1 wx 2 , where w is a red vertex (see Figure 3 (a)). Then int(C) contains none of black vertices, red vertices and black edges in F ; in this sense we also say that int(C) is "empty".
Proof. By the definition of D × W , the proposition follows directly from the fact that the drawing of edge e w is most near one side of the 2-path Figure 3: Some 3-cycles and 2-cycles in F .

Proposition 2
Assume that H has no separating 2-cycles. If C is a 2-cycle in H that consists of two multiple edges e 1 and e 2 joining two black vertices x 1 and x 2 (see Figure 3 (b)), then either int(C) or ext(C) contains neither black vertices nor red vertices.
Proof. As H has no separating 2-cycles, either int(C) or ext(C) contains no black vertices. Assume that int(C) does not contain black vertices.
Suppose that int(C) contains red vertices. Then, int(C) contains white vertices of D × . As int(C) does not contain black vertices, each white vertex in int(C) is of degree at most 2 in D × . Thus, we can redraw the edges of D in int(C) such that these edges make no crossings, and then obtain a 1-planar drawing of G with fewer crossings than D, contradicting to the choice of D. Hence int(C) does not contain red vertices and the conclusion holds. ✷

Proposition 3
Assume that H contains no separating 2-cycles. Then the edge multiplicity of H is at most 2.
Proof. Assume to contrary that H has three multiple edges e 1 , e 2 and e 3 which join the same pair of black vertices x 1 and x 2 . Then these three edges divide the plane into three regions, denoted by α, β and γ, as shown in Figure 4 (a). By Proposition 2, at least two of these three regions contain neither red vertices nor black vertices, except on its boundary. We may assume α and γ are such two regions.
Let P = x 1 wx 2 be the 2-path of F that corresponds to edge e 3 , where w is a red vertex. Thus, this path must be within region β, as shown in Figure 4 (b).
As P is within region β, black edges e 1 and e 2 are in different sets int(e 3 ∪ P ) and ext(e 3 ∪ P ), a contradiction to Proposition 1. The proof is then completed. ✷  Let C be a cycle and P be a path in H such that the end vertices of P are the only vertices in both C and P . When we say that P lies in int(C) (resp. ext(C)), it means that all edges and internal vertices of P lie in int(C) (resp. ext(C)).

Proposition 4
Assume that H has no separating 2-cycles. Let C be a 3-cycle of H consisting of black vertices x 1 , x 2 and x 3 , and e be the edge on C joining x 1 and x 3 . Assume that e ′ is a partnered edge in H which is parallel to e. If P = x 1 wx 3 and P ′ = x 1 w ′ x 3 are the 2-paths in F corresponding to e and e ′ respectively, then one of P and P ′ lies in int(C) and the other in ext(C).
Proof. Let C 0 denote the 2-cycle of H consisting of edges e and e ′ . By Proposition 2, we may assume that int(C 0 ) contains neither black vertices nor red vertices. Thus, both w and w ′ are in ext(C 0 ).
Let C 1 denote the 3-cycle of F consisting of edge e and path P and C ′ 1 the 3-cycle of F consisting of edge e ′ and path P ′ . By Proposition 1, both int(C 1 ) and int(C ′ 1 ) are empty. Thus, the subgraph F [{x 1 , x 3 , w, w ′ }] is as shown in Figure 5 (a).
As these three regions int(C 0 ), int(C 1 ) and int(C ′ 1 ) do not contain black vertices, x 2 must be in ext(C 2 ), where C 2 is the 4-cycle of F consisting of paths P = x 1 wx 3 and P ′ = x 1 w ′ x 3 . As F is a plane graph, path x 1 x 2 x 3 must lies in ext(C 2 ), as shown in Figure 5  Proof. Let e 1 , e 2 and e 3 be the three edges on C. Suppose that e i is not a simple edge of H, where 1 ≤ i ≤ 3. Then e i is parallel to another partnered edge e ′ i of H. Let P i and P ′ i be 2-paths in F which correspond to edges e i and e ′ i respectively. Since H has no separating 2-cycles, by Proposition 4, int(C) contains a red vertex that is on P i or P ′ i .
The above conclusion implies that the number of red vertices in int(C) is not less than the number of partnered edges on C. Thus, the result holds. ✷

Some lemmas
Let G be a bipartite graph with partite sets X and Y and O be a disk on the plane. If D is a 1-planar drawing of G that draws all vertices of X on the boundary of O and all vertices of Y and all edges of G in the interior of O, then we say that D is a 1-disc O X drawing of G.
Lemma 1 Let G be a bipartite graph with partite sets X and Y , and let D be a 1-disc O X drawing of G with the minimum number of crossings k. If |X| = 3, then k ∈ {0, 1, 3} and |E(G)| ≤ 2|Y | + 1 + √ k , i.e., x 3 x 1 Proof. Assume that |X| = 3. For any integer i ≥ 0, let Y i be the set of vertices y in Y with d G (y) = i. As |X| = 3 and Y is independent in G, Y i = ∅ holds for all i ≥ 4.
It can be checked easily that, for each vertex y in ∈ Y , if y / ∈ Y 3 , then y is not incident with any crossed edge. Thus, G − i≤2 Y i has exactly k crossings, and it suffices to show that |Y 3 | ≤ 3 and The rest of the proof will be completed by showing the following claims.
Suppose that |Y 3 | ≥ 4. Then, there exists a bipartite 1-planar drawing D ′ isomorphic to K 3,2|Y 3 | obtained from D[X ∪ Y 3 ] by copying all vertices and edges in the interior of O X to its exterior, implying that K 3,8 is 1-planar. It is a contradiction to the fact that K 3,7 is not 1-planar due to Czap and Hudák [3].
Claim (b) can be verified easily.
Claim (c): For any two vertices y 1 , y 2 ∈ Y 3 , some edge incident with y 1 crosses with some edge incident with y 2 , as shown in Figure 6 (a).
If Claim (c) fails, then G[X ∪ {y 1 , y 2 }] is a plane graph and we can get a drawing of K 3,3 from G[X ∪ {y 1 , y 2 }] by adding a new vertex y ′ and three edges joining y ′ to all vertices in X in the exterior of O X without any crossing, implying that K 3,3 is planar, a contradiction.
If c = 1, since G is simple, each face of H is a cellular face and has size at least 3. Then, in this case, the conclusion can be proved easily by applying the Euler's formula. Now we assume that c ≥ 2. We can obtain a simple and connected plane graph G ′ from G by adding c − 1 edges.
For every noncellular face F of G, we assume that its boundary consists of ℓ disjoint closed walks of G, and then we can add ℓ − 1 new edges (not add the vertex) by appropriately drawing these new edges within F so that F is transformed into a cellular face of size at least 4 because |V (G)| ≥ 3. Therefore, the resulting graph G ′ is a simple and connected plane, and all faces of G ′ are cellular.
Note that adding the c − 1 new edges does not produce new cellular 3-faces, and thus G ′ has exactly t faces of size 3. The conclusion for connected plane graphs implies that Let G be a simple and bipartite plane graph with |V (G)| ≥ 3. If G has exactly c components and t cellular faces whose boundaries are of length at least 6, Proof. If G is connected (i.e. c = 1), since G is bipartite and simple, then each face of G is a cellular face, and has the size at least 4. Because G has t faces of size at least 6, it follows from the Euler's formula that |E(G)| ≤ 2|V (G)| − 4 − t.
Now assume that c ≥ 2. We can obtain a simple and connected bipartite plane graph G ′ from G by adding c − 1 edges.
For every noncellular face F of G consisting of ℓ distinct closed walks, similar to the proof of Lemma 2, we can add ℓ − 1 new edges within this noncellular face so that F is transformed into a cellular face. We can ensure that those new added edges join the vertices in different partite sets of G. Hence the resulting plane graph G ′ is simple, bipartite and connected. Clearly, all faces of G ′ are cellular, and G ′ has at least t faces whose boundaries are of length at least 6. The conclusion for bipartite and connected plane graphs implies that As V (G ′ ) = V (G) and |E(G ′ )| = |E(G)| + c − 1, the above inequality implies that

Proof of Theorem 3
The whole section contributes to the proof of Theorem 3.
We will prove the following claims to show that this assumption leads to a contradiction.
By the assumption of χ, we have χ ≥ 4. ✷ In the following, we assume that G is a bipartite 1-planar graph with bipartitions X and Y , where χ = |X| ≤ |Y |, such that Let D be a 1-planar drawing of G with the minimum number of crossings and W be the set of its crossings. Introduced in Section 3, D × W is a plane graph extended from D × , and F and H are the subgraphs D × W [X ∪W ] and D × W [X] of D × W respectively. All vertices in X are black vertices, all vertices in Y are white vertices and all vertices in W are red vertices.
We are now going to prove the following claim. Proof. Assume to the contrary that H has a separating 2-cycle C consisting of two parallel edges e 1 and e 2 joining black vertices x 1 and x 2 (see Figure 3 (b)), such that both int(C) and ext(C) contain black vertices.
Thus, Claim 4 holds. ✷ Proof. Since H ′ is a simple plane graph with exactly t ≥ 1 cellular 3-faces and |V (H ′ )| = |X| = χ ≥ 4, by Lemma 2, On the other hand, by Proposition 5, for any j ∈ {0, 1, 3} and 1 ≤ i ≤ t j , at least 3−j edges on the boundary of ∆ We see that the graph G ′ is a bipartite 1-planar graph with a bipartition X and and As the number of crossings of D equals to |E(H)| and D ′ has no crossings lying in the interior of any cellular 3-face of H ′ , D ′ has exactly |E(H)| − (t 1 + 3t 3 ) crossings.
Proof. Note that the simple and bipartite plane graph G * is obtained from G by removing all white vertices and edges of G lying in the interiors of all cellular 3-faces of H ′ and, for each crossing of D not lying in any cellular 3-face of H ′ , removing exactly one edge of G involved in this crossing.
Because the edges of H (and thus H ′ ) are not crossed with the edges of G (and thus G * ), we observe that G * * is also a simple and bipartite plane graph and has at least t cellular 6-faces. Applying Lemma 3 to G * * yields that Then, (11) implies that |E(G * )| ≤ 2|V (G * )| − 4 − t. This proves the claim. ✷ Proof. By (9), we have Then, by Claims 4, 5 and 6, ✷ Clearly, Claim 7 contradicts the assumption in (1). Hence Theorem 3 holds. ✷

Remarks
For any x ≥ 3 and y ≥ 6x − 12, Czap, Przybylo andSkrabuláková [5, Lemma 4] constructed a bipartite 1-planar graph G with partite sets X and Y such that |E(G)| = 2|V (G)| + 4|X| − 12. Notice that the 1-planar drawing D of this graph G given in [5] has the following property: (*) each vertex in X is incident with crossed edges in D.
The proof of Theorem 3 also yields that, if |E(G)| = 2|V (G)| + 4|X| − 12 holds for a bipartite 1-planar graph G with partite sets X and Y , where 4 ≤ |X| ≤ |Y |, and D is a 1-planar drawing of G with the minimum number of crossings, then the graph H ′ introduced in Section 3 does not have isolated vertices, i.e., property (*) above holds.
Based on the above observations, we propose the following problem. Problem 2 Let G be a bipartite 1-planar graph with partite sets X and Y , where 4 ≤ |X| ≤ |Y |. If D is a 1-planar drawing of G with the minimum number of crossings and |X >0 | ≥ 3, where X >0 is the set of vertices in X which are incident with crossed edges of D, does |E(G)| ≤ 2|V (G)| + 4|X >0 | − 12 hold?
Theorem 3 shows that |E(G)| ≤ 2|V (G)| + 4x − 12 holds for any bipartite 1-planar graph G with bipartite sets of sizes x and y, where 2 ≤ x ≤ y. For any x ≥ 3 and y ≥ 6x − 12, Czap, Przybylo andSkrabuláková [5] constructed a bipartite 1-planar graph G with bipartite sets of sizes x and y and |E(G)| = 2|V (G)| + 4x − 12. Notice that these graphs constructed in [5] have minimum degree 3. By Theorem 1, any bipartite 1-planar graph of n vertices has at most 3n − 8 edges, implying that its minimum degree is at most 5. We wonder if the result in Theorem 3 can be improved for bipartite 1-planar graphs with higher minimum degrees or connectivity.
Let t ≥ 2. A drawing of a graph is t-planar if each of its edges is crossed at most t times. If a graph has a t-planar drawing, then it is t-planar. Does Theorem 3 have an analogous result for bipartite 2-planar graphs?
Problem 4 Let G be a bipartite 2-planar graph with partite sets X and Y , where 2 ≤ |X| ≤ |Y |. Determine constants a, b and c such that |E(G)| ≤ a|V (G)|+b|X|+c.
Lemma 1 gives an upper bound for the size of a bipartite graph G with partite sets X and Y , where |X| = 3, which has a 1-disc O X drawing. Can this result be generalized for such a bipartite graph without the condition that |X| = 3?
Problem 5 Let G be a bipartite graph with partite sets X and Y which has a 1-disc O X drawing. Is it true that |E(G)| ≤ 2|Y | + 5|X|/3 − 2?