An Improved Upper Bound on the Queue Number of Planar Graphs

A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+1)$$\end{document}-rainbows, that is, k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document} independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, which is defined as the minimum value of k for which a k-queue layout is feasible. Recently, Dujmović et al. [J. ACM, 67(4), 22:1–38, 2020] showed that the queue number of planar graphs is at most 49, thus settling in the positive a long-standing conjecture by Heath, Leighton and Rosenberg. To achieve this breakthrough result, their approach involves three different techniques: (1) an algorithm to obtain 2-queue layouts of outerplanar graphs, (2) an algorithm to obtain 5-queue layouts of planar 3-trees, and (3) a decomposition of a planar graph into so-called tripods. In this work, we push further each of these techniques to obtain the first non-trivial improvement of the upper bound on the queue number of planar graphs from 49 to 42\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$42 $$\end{document}.


Introduction
Linear layouts of graphs have a long tradition of study in different contexts, including graph theory and graph drawing, as they form a framework for defining different graph-theoretic parameters with several applications; see, e.g., [1]. In this regard, one seeks to find a total order of the vertices of a graph that reaches a certain optimization goal [2][3][4]. In this work, we focus on a well-studied type of linear layouts, called queue layout [5][6][7][8], in which the goal is to minimize the size of the largest rainbow, namely, a set of independent edges that are pairwise nested. Equivalently, the problem asks for a linear order of the vertices and a partition of the edges into a minimum number of queues (called queue number), such that no two independent edges in the same queue are nested [7]; see Fig. 1b.
Queue layouts of graphs were introduced by Heath and Rosenberg [7] in 1992 as the counterpart of stack layouts (widely known also as book embeddings), in which the edges must be partitioned into a minimum number of stacks (called stack number), such that no two edges in the same stack cross [9]; see Fig. 1c. Since their introduction, queue layouts of graphs have been a fruitful subject of intense research with several important algorithmic and combinatorial results proposed over the years [6,8,[10][11][12][13][14]; for an introduction, we refer the reader to [15].
The most intriguing problem in this research field is undoubtedly the problem of specifying the queue number of planar graphs, that is, the maximum queue number over all planar graphs. This problem dates back to 1992, when Heath, Leighton and Rosenberg conjectured that the queue number of planar graphs is bounded [6]. Notably, despite the different efforts [11,16,17], this conjecture remained unanswered for more than two decades. That only changed in 2019 with a breakthrough result of Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [5,18], who managed to settle in the positive the conjecture, as they showed that the queue number of planar graphs is at most 49. The best-known corresponding lower bound is 4 due to Alam et al. [10].
It is immediate to see, however, that the gap between the currently best-known lower and upper bounds is rather large, which implies that the exact queue number of planar graphs is, up to the point of writing, still unknown. Note that this is in contrast with the stack number of planar graphs, which was recently shown to be exactly 4 [19,20]. Also, the existing gap in the bounds on the queue number of planar graphs gives the intuition that it is unlikely the upper bound of 49 by Dujmović at al. [5] is tight, even though in the last two years no improvement appeared in the literature.

Our Contribution
We verify the aforementioned intuition by reducing the upper bound on the queue number of planar graphs from 49 to 42 (see Theorem 7 in Sect. 5). This improvement requires non-trivial adjustments to each of the three algorithms that are involved in the approach by Dujmović at al. [5], namely: (i) an algorithm to obtain 2-queue layouts of outerplanar graphs by first computing straight-line drawings of them in which the ydistance of any two adjacent vertices is either 1 or 2 [21]; see Sect. 4.1, (ii) an algorithm to obtain 5-queue layouts of planar 3-trees [10]; see Sect. 4.2, and (iii) a decomposition of a planar graph into so-called tripods [5]; see Sect. 4.3. Although we assume familiarity with these algorithms, in Sect. 3 we outline their most important aspects. For preliminary notions and standard terminology refer to Sect. 2.

Preliminaries
A drawing of a graph maps each vertex to a distinct point of the Euclidean plane and each edge to a Jordan curve connecting its endpoints. A drawing of a graph is planar, if no two edges cross (except at common endpoints). A graph that admits a planar drawing is called planar; see Fig. 1a. A planar drawing partitions the plane into topologically connected regions called faces; its unbounded one is called outer face. A combinatorial embedding of a planar graph is an equivalence class of planar drawings that are pairwise topologically equivalent, that is, they define the same set of faces up to an orientation-preserving homeomorphism of the plane. A planar graph together with a combinatorial embedding is a plane graph. A graph that admits a planar drawing, in which all vertices are incident to its outer face, is called outerplanar. It is known that outerplanar graphs have treewidth at most 2. A planar graph with treewidth at most 3 is commonly referred to as a planar 3-tree. Planar 3-trees are subgraphs of maximal planar 3-trees, which are recursively defined as follows. In the base case, the graph is a 3-cycle. Otherwise, there exists a vertex of degree 3 whose removal yields a maximal planar 3-tree with one vertex less. A linear layout of a graph is defined by a total order of its vertices and an objective over its edges. We denote the (total) vertex order of a simple undirected graph G by ≺, such that for any two vertices u and v of G, u ≺ v if and only if u precedes v in the order. Let F be a set of k ≥ 2 independent edges Fig. 2. Finally, two independent edges that form a 2-rainbow (2-necklace, 2-twist) are referred to as nested (disjoint, crossing, respectively).
There are several types of linear layouts of graphs in the literature, with queue layouts being among the most studied of them. Formally, a k-queue layout of a graph G consists of a vertex order ≺ of G and a partition of the edges of G into k sets of pairwise non-nested edges with respect to ≺, called queues. A preliminary result by Heath and Rosenberg [7] states that a graph admits a k-queue layout if and only if it admits a vertex order in which no (k + 1)-rainbow is formed. The queue number of a Illustration of a a 3-rainbow, b a 3-necklace, and c a 3-twist (a) (b) Fig. 3 Introducing a degree-2 vertex v along the upper envelope, when its two neighbors u and w are connected with a a span-1 edge, and b a span-2 edge graph G, denoted by qn(G), is the minimum k, such that G admits a k-queue layout. Accordingly, the queue number of a class of graphs is the maximum queue number over all its members.

Outline of the Involved Techniques
In this section, we summarize the main aspects of the three algorithms mentioned in Sect. 1 that are involved in the approach by Dujmović et al. [5] to achieve the bound of 49 on the queue number of planar graphs. The first is an algorithm by Dujmović, Pór, and Wood [21] to compute 2-queue layouts of outerplanar graphs (Sect. 3.1). The second one is by Alam et al. [10] to compute 5-queue layouts of planar 3-trees (Sect. 3.2). The last one is the actual algorithm by Dujmović et al. [5] to compute 49-queue layouts of planar graphs (Sect. 3.3).

Outerplanar Graphs
As already stated, the main ingredient of the algorithm by Dujmović, Pór and Wood [21] is the construction of a straight-line drawing (G) of a maximal outerplane graph G that can be transformed into a 2-queue layout of G. The recursive construction of (G) maintains the following invariant properties: To obtain drawing (G) of G, vertex v is inserted into (G ) as follows. Let u and w be the neighbors of v in G. Since G is maximal outerplane, (u, w) is an edge of G that lies on the outer face of (G ). By Invariant (O.2), (u, w) is either a span-1 or a span-2 edge. Assume that (u, w) is along the upper envelope of (G ); the case where it is along the lower envelope is symmetric. Assume first that (u, w) has span 1 in (G ) and without loss of generality that y(u) = y(w) + 1. Then, vertex v is placed such that , and (ii) the edges assigned to the first (second) queue in the layout are those with span 1 (span 2, respectively) in (G).

Planar 3-trees
Alam et al. [10] adopt a peeling-into-levels approach [22] to produce a 5-queue layout of a maximal plane 3-tree H . Initially, the vertices of H are partitioned into levels L 0 , . . . , L λ with λ ≥ 1, such that (T.2) Vertices of each connected component of level L i appear consecutively in ≺ H for every i = 0, . . . , λ; (T.3) All level edges of H are assigned to two queues denoted by Q 0 and Q 1 ; (T.4) For every i = 0, . . . , λ − 1, the binding edges between vertices of L i and L i+1 are assigned to three queues Q 2 , Q 3 and Q 4 as follows. For each connected component c of H i+1 , let x, y and z be its three neighbors in H i so that x ≺ H y ≺ H z. Then, the binding edges between L i and L i+1 with one endpoint in c are assigned to Q 2 , Q 3 or Q 4 if their other endpoint is x, y or z, respectively.

General Planar Graphs
Dujmović et al. [5] present an algorithm to produce a 49-queue layout of a (general) plane graph. Central in their algorithm is the notion of an H -partition 2 , defined as follows. Given a graph G, an H -partition of G is a partition of the vertices of G into . .) of its vertices according to their distance from a specific vertex of G. In other words, it is a special type of H -partition, where H is a path and each bag V i corresponds to a layer. In this special type of H -partition, an intra-bag edge is called intra-layer, while an inter-bag edge is called inter-layer 3 .
An H -partition has layered-width with respect to a BFS-layering L if each bag of H has at most vertices on each layer of L.
The following lemma is the main tool of the algorithm by Dujmović et al. [5], since it yields a construction of a queue-layout of a graph G, whose number of queues depends on the layered-width of an H -partition of G and the queue number of H . Lemma 1 (Dujmović et al. [5]) For all graphs G and H , if H admits a k-queue layout and G has an H -partition of layered-width with respect to some layering In the original proof of Lemma 1 (see [5,Lemma 9]), the order of the vertices of G on each layer of L is defined as follows. Let x 1 , . . . , x h be the vertices of H as they appear in a k-queue layout QL(H ) of H , that is, where each subset A x j ∩ V i is ordered arbitrarily, for j = 1, . . . , h. This gives the total order ≺ G for the vertices of G. The edge-to-queue assignment, which completes the construction of the queue layout QL(G) of G, exploits the following two properties: Recall that the edges of G are classified into four categories given by the bags of the Hpartition and the layers of L, namely, intra-layer intra-bag edges, inter-layer intra-bag edges, intra-layer inter-bag edges and inter-layer inter-bag edges. The edge-to-queue assignment is based on these four categories. We start with the intra-bag edges of G, that is, edges whose endpoints belong to the same bag; see Fig. 4a.
(E.1) Intra-layer intra-bag edges of G are assigned to at most 2 queues, as the queue number of K is 2 [22]; see red edges in Fig. 4a. (E.2) Inter-layer intra-bag edges of G are assigned to at most queues, as the queue number of K , is , when all vertices of the first bipartition precede those of the second; see blue edges in Fig. 4a.
The remaining edges of G are the inter-bag edges that connect vertices of different bags; see Fig. 4b. Inter-layer inter-bag edges are further partitioned into two (E.4) Forward inter-layer inter-bag edges of G are assigned to at most k queues; for two consecutive layers, an edge of H corresponds to a subgraph of K , , where all vertices of the first bipartition precede those of the second; see purple edges of Fig. 4b. (E.5) Symmetrically all backward inter-layer inter-bag edges of G are assigned to at most k queues; see orange edges in Fig. 4b.
The next property follows from the edge-to-queue assignment of inter-bag edges of G. Details can be found in the proof of Lemma 9 in [5]. In order to bound the parameters k and used in Lemma 1, the fact that G is maximal plane is being exploited. To this end, a few more ingredients are needed. A vertical path of G in a BFS-layering L is a path P = v 0 , . . . , v κ of G consisting only of edges of the BFS-tree of L and such that if v 0 belongs to V i in L, then v j belongs to V i+ j , with j = 1, . . . , κ. Further, we say that v 0 and v κ are the first and last vertices of P. A tripod of G consists of up to three pairwise vertex-disjoint vertical paths in L whose last vertices form a clique of size at most 3 in G. We refer to this clique as the base of the tripod. Dujmović et al. [5] describe a recursive algorithm that partitions the vertices of G into tripods, whose bases are triangular faces of G. In particular, they show that for any BFS-layering L, G admits an H -partition with the following properties: (P.4) H is a planar 3-tree and thus QL(H ) is a k-queue layout with k ≤ 5 [10]. (P.5) Its layered-width is at most 3, since each bag induces a tripod in G.

Refinements of the Involved Techniques
In this section, we present refinements of the algorithms outlined in Sect. 3 that will allow us to reduce the upper bound on the queue number of planar graphs from 49 to 42.

Outerplanar Graphs
The initial algorithm by Dujmović et al. [21], outlined in Sect To maintain Invariant (O.3), we observe that a biconnected maximal outerplane graph G with at least four vertices, contains at least two non-adjacent degree-2 vertices. Let x be such a degree-2 vertex of G, which we assume to be fixed in the recursive construction of (G). In particular, by our previous observation, we can always remove a degree-2 vertex that is different from x at every recursive step. This ensures that x will eventually appear in the triangle T that is drawn at the base of the recursion. We draw T , such that x is its bottommost vertex while its two incident edges are of span-1 and span-2 as follows. Let y and z be the other two vertices of T . We draw x, y and z at points (2, 0), (0, 1) and (1, 2), respectively. This implies that (x, y) forms the lower envelope of T , while (y, z) and Proof For (a), consider a vertex u of G. If u is the top vertex of a face, then u is incident to a span-2 edge (u, v) with y(u) > y(v). By Invariant (O.3), u is a successor of v in the recursive approach by Dujmović et al. [21], namely, when u is placed in the recursively constructed drawing, vertex v belongs to its upper envelope. We now claim that u cannot be incident to two span-2 edges (u, v) and (u, v ) with y(u) > y(v) and y(u) > y(v ), that is, with the properties mentioned above; this claim implies part (a) of the lemma. Assuming the contrary, by Invariant (O.2), when u is placed in the drawing at most one edge incident to u has span 2. So, at most one of (u, v) and (u, v ) is drawn when u is placed in the drawing, which implies that at least one of v or v , say v , is a successor of u. Thus, y(u) < y(v ) holds; a contradiction.
For (b), assume for a contradiction that G contains a vertex u, which is the middle vertex of three triangular faces, say T 1 , T 2 and T 3 , of (G). For a significantly small constant ε > 0, let p 1 = (x(u) − ε, y(u)) and p 2 = (x(u) + ε, y(u)), and consider the two horizontal rays r 1 and r 2 emanating from vertex u, such that r 1 and r 2 contain points p 1 and p 2 , respectively. Since u is middle vertex for T 1 , T 2 and T 3 , it follows by Invariant (O.2) that the span-2 edge of each of T 1 , T 2 and T 3 crosses either r 1 or r 2 ; by the choice of ε, we may assume that if a span-2 edge of one of the faces T 1 , T 2 and T 3 crosses r 1 (r 2 ), then point p 1 ( p 2 , respectively) lies in the interior of this face. On the other hand, by outerplanarity, each of the points p 1 and p 2 can be contained in at most one of T 1 , T 2 and T 3 ; a contradiction.
Note that part (a) of Lemma 2 requires Invariant (O.3), that is, it does not necessarily hold for all drawings obtained by the algorithm by Dujmović et al. [21]. On the other hand, part (b) of Lemma 2 is a property of the original algorithm as it holds for drawings that do not necessarily satisfy Invariant (O.3).

Planar 3-trees
To maintain Invariant (T.3), Alam et al. [10] use the algorithm by Dujmović et al. [21] to assign the level edges of L 0 , . . . , L λ of the input plane 3-tree H to two queues Q 0 and Q 1 , since on each level these edges induce a (not necessarily connected) outerplane graph. In our approach we adopt the modification for the algorithm by Dujmović et al. [21] introduced in Sect. 4. To maintain Invariant (T.4), Alam et al. [10] adopt the following assignment scheme for the binding edges between L j and L j+1 to queues Q 2 , Q 3 and Q 4 , for each j = 0, . . . , λ − 1. Consider a binding edge (u, v) with u ∈ L j and v ∈ L j+1 . Then, vertex u belongs to a connected component C u of the subgraph H j of H induced by the level-L j vertices. Similarly, vertex v belongs to a connected component C v of the corresponding subgraph H j+1 of H . Furthermore, C u is outerplane and its 2-queue layout has been computed by the algorithm by Dujmović et al. [21], while C v resides in the interior of a triangular face T v of C u in the embedding of H , such that u is on the boundary of T v . In the edge-to-queue assignment scheme by Alam et al. [10], edge (u, v) is assigned to Q 2 , Q 3 or Q 4 if and only if u is the middle, top or bottom vertex of T v , respectively [10]. For a vertex u ∈ L j with j = 0, . . . , λ − 1, we denote by N 2 (u), N 3 (u) and N 4 (u) the neighbors of u in L j+1 such that the edges connecting them to u are assigned to queues Q 2 , Q 3 and Q 4 , respectively. The next property follows from Lemma 2.

Lemma 3
For j = 0, . . . , λ − 1, let u ∈ L j be a vertex of H in our modification of the peeling-into-levels approach by Alam et al. [10]. Then, the vertices of N 2 (u) precede those of N 3 (u) in ≺ H . Also, the vertices of N 2 (u) belong to at most two connected components of H j+1 (residing within distinct faces of H j ); the same holds for the vertices of N 3 (u).
Proof The first part of the lemma is proven in [10], although it is not explicitly stated as part of Invariants (T.1)-(T.4). For the second part, consider a binding edge (u, v) with v ∈ N 2 (u); a similar argument applies when v ∈ N 3 (u). By the definition of

General Planar Graphs
Dujmović et al. [5] recursively compute the bags (that is, the tripods) of the H -partition. In this recursive approach, each newly discovered tripod τ is adjacent to at most three other tripods τ 1 , τ 2 and τ 3 already discovered. We say that τ 1 , τ 2 and τ 3 are the parents of τ ; see Fig. 5a. Also, each non-empty vertical path of τ is connected to only one of its parents via an edge of the BFS-tree used to construct the BFS-layering L (black in Fig. 5a). This property gives rise to at most three sub-instances (gray in Fig. 5a), which are processed recursively to compute the final tripod decomposition. Note that, in general, one or more vertical paths of a tripod may have no vertices; see Fig. 5b. We refer to such a tripod as degenerated. In addition, we call a degenerated tripod empty if it shares a vertex with each of its parents. Hence, empty tripods have no vertices and as a result do not contribute to H . To avoid considering degenerated tripods that are not empty explicitly, we introduce an appropriate augmentation of the input graph. More precisely, let f be the base triangle of a non-empty degenerated tripod τ , and let T be the BFS tree that was used to compute τ . We augment G by adding a triangle f in the interior of face f . By an appropriate triangulation, we can obtain a triangulated plane graph G that contains G as a subgraph, such that all vertices of f are leaves of T , which gives rise to a BFS tree T of G . In this way, face f of G gets three leaves of T inside it, one attached to each vertex of f . Hence, the base triangle of the corresponding tripod in G is not degenerated. This allows us to assume without loss of generality in the following that, in the tripod decomposition, either each tripod is empty or non-degenerated, that is, no vertical path of it is empty.
For i = 1, 2, 3, let p 1 i , p 2 i and p 3 i be the three vertical paths of tripod τ i . Up to renaming, we assume that τ lies in the cycle bounded by (parts of) p 1 1 , p 2 1 , p 1 2 , p 2 2 , p 1 Note that Properties (P.6), (P.7), and (P.8) hold even if τ has less than three parents, or if the cycle bounding the region of τ does not contain two vertical paths of each parent tripod.
In the original algorithm by Dujmović et al. [5], each vertex v τ in H corresponds to a non-empty tripod τ in G, and an edge (v τ , v τ ) exists in H , if τ is a parent of τ in G, or vice versa. By construction, H is a connected partial planar 3-tree, which is arbitrarily augmented to a maximal planar 3-tree H . Then, the algorithm by Alam et al. [10] is employed to compute a 5-queue layout of H as required by Lemma 1. Here, we adopt a particular augmentation to guarantee an additional property for the graph H (see Lemma 4). Similarly to the original approach, we contract the vertices of each non-empty tripod τ of G to a single vertex v τ . However, in our modification, we keep self-loops that occur when an edge of G has both endpoints in τ (unless this edge belongs to one of its vertical paths), as well as, parallel edges that occur when two vertices of τ have a common neighbor not in τ . Two important properties of this contraction scheme that follow directly from planarity are given below; see Note that the aforementioned contraction scheme creates self-loops and pairs of parallel edges. We focus on homotopic self-loops and pairs of parallel edges, which contain no vertex either in the interior or in the exterior region that they define. We remove such self-loops and keep one copy of such parallel edges. Then, we subdivide each (non-homotopic) self-loop twice, and for each edge with multiplicity m > 1, we subdivide all but one of its copies. In this way, each vertex v τ corresponding to a tripod τ in G always lies in the interior of a separating 3-cycle C that contains all the vertices corresponding to its parent tripods on its boundary. To see this, observe that if τ has three parent τ 1 , τ 2 and τ 3 , then C is formed by v τ 1 , v τ 2 and v τ 3 , such that v τ is (a) (b) Fig. 7 a A tripod τ in G where the edges incident to its three vertical paths are drawn dotted, and b the result after contracting τ to v τ connected to each of them. If τ has two parent τ 1 and τ 2 , then C is formed by v τ 1 , v τ 2 and a subdivision vertex, such that v τ is connected to v τ 1 and v τ 2 . Finally, if τ has only one parent τ 1 , then C is formed by v τ 1 and two subdivision vertices, such that v τ is connected to v τ 1 . Since subdivision vertices are of degree 2, the result is a simple (possibly not maximal) planar 3-tree, which is a supergraph of H . To derive H , we augment it to maximal by adding edges, while maintaining its embedding [23]. We are now ready to state the additional property that H has.

Lemma 4 Let v τ and v τ p be two vertices of H that correspond to a tripod τ and to a parent tripod τ p of τ in G. If L i and L j are the levels of v τ and v τ p in the peeling-into-levels approach for H , then i ≥ j.
Proof Let C be the inclusion-minimal separating 3-cycle of H containing v τ in its interior and all vertices that correspond to the parent tripods of τ on its boundary. Let L k , L l and L m be the levels of the three vertices of C, with k ≤ l ≤ m, in the peeling-into-levels approach for H . As τ p is a parent of τ , j ∈ {k, l, m} holds. Since C is a 3-cycle and since each edge in the peeling-into-level approach is either level or binding, m ≤ k + 1 holds. The fact that v τ lies in the interior of C and is connected to each of the vertices in H corresponding to its parent tripods in G, implies that v τ is on level L k+1 , that is, i = k + 1. So, j ≤ m ≤ k + 1 = i holds.
As in the original algorithm by Dujmović et al. [5], we compute a 5-queue layout QL(H ) of H . However, we use our modification of the algorithm by Alam et al.
where A x 1 , . . . , A x h are the bags, that is, the tripods, of the H -partition of G. Unlike in the original algorithm, we do not order the vertices in each subset A x j ∩ V i , with j ∈ {1, . . . , h}, arbitrarily. Instead, we carefully choose their order; we describe this choice in the rest of this section. Let τ be the tripod of bag A x j . Then, A x j ∩ V i contains  Fig. 8b. Hence, the cyclic order of the edges around v τ in H is not the same as the cyclic order of the edges incident to vertices of τ along a clockwise traversal of τ in G, contradicting Property (P.9). Hence, Case (i) and (ii) cannot apply simultaneously, as we initially claimed.

Reducing the Bound
To reduce the upper bound on the queue number of planar graphs, we turn our attention to the analysis of the required number of queues for the intra-bag inter-layer edges (E.2), as well as for the inter-bag edges (that is, either intra-layer (E.3) or interlayer (E.4)-(E.5)); refer to Section 3.3.
For intra-bag inter-layer edges (E.2), the original algorithm by Dujmović et al. [5] uses three queues, since the layered width is 3; see blue edges in Fig. 4a. We prove that no three intra-bag inter-layer edges form a 3-rainbow, implying that the upper bound on the queue number of planar graphs can be improved from 49 to 48. Note that this first improvement does not require any of the modifications presented in Sect. 4.

Lemma 5
In the queue layout computed by the algorithm by Dujmović et al. [5], no three intra-bag inter-layer edges of G form a 3-rainbow.
Proof Assume to the contrary that there exist three such edges (u 1 , v 1 ), (u 2 , v 2 ) and (u 3 , v 3 ) forming a 3-rainbow in QL(G) such that By Property (P.1) these edges belong to the same bag A of the H -partition, while by Property (P.2) their endpoints belong to two consecutive layers V i and V i+1 of L, for some i ≥ 0. The underlying linear order implies that vertices u 1 , u 2 and u 3 belong to V i , while v 1 , v 2 and v 3 belong to V i+1 . The order of A ∩ V i and A ∩ V i+1 is u 1 ≺ G u 2 ≺ G u 3 and v 3 ≺ G v 2 ≺ G v 1 ; see Fig. 8c. Let p 1 , p 2 and p 3 be the first, second and third vertical paths of tripod τ forming A. Then, (u 1 , v 3 ) ∈ p 1 , (u 2 , v 2 ) ∈ p 2 and (u 3 , v 1 ) ∈ p 3 . However, (u 1 , v 1 ) and (u 3 , v 3 ) contradict Property (P.8). Hence, there is no 3-rainbow formed by intra-bag inter-layer edges of G.
For inter-bag edges (E.3)-(E.5), the algorithm by Dujmović et al. [5] uses 3·15 queues, since the layered width is 3 and the best known upper bound on the queue number k of planar 3-trees is 5; see green, purple and orange edges in Fig. 4b. We exploit Property (P.3) to prove that 3 · 13 queues suffice for inter-bag edges. This further improves the upper bound on the queue number of planar graphs from 48 to 42. Note that this proof is more involved than the previous one and requires all modifications presented in Sect. 4.

Lemma 6
In the queue layout computed by our modification of the algorithm by Dujmović et al. [5], the inter-bag edges of G do not form a 40-rainbow.
Proof To prove the statement, it suffices to show that the inter-bag edges of G that belong to each of (E.3), (E.4) and (E.5) form at most a 13-rainbow. We focus on the edges of (E.3), that is, on the intra-layer inter-bag edges of G; a similar argument applies for the other two types of edges, that is, forward and backward inter-layer inter-bag edges. We partition the edges of (E.3) into five parts E 0 3 , . . . , E 4 3 , such that for i = 0, . . . , 4, part E i 3 contains the following type-(E.3) edges of G: By Property (P.3), and since = 3, the edges of E i 3 can form an r -rainbow with r ≤ 3 (this property actually implies the initial bound of 3 · 5 = 15 queues for all intra-layer inter-bag edges of G by Dujmović et al. [5]). We next prove that neither the edges of E 2 3 nor the edges of E 3 3 form a 3-rainbow, which yields the desired reduction from 15 to 13 on the size of the maximum rainbow formed by all edges of (E.3). Assume that this is not true, say for the edges of E 2 3 ; a symmetric argument applies for the edges of , that is, intra-layer edges, their endpoints u 1 , u 2 , u 3 , v 1 , v 2 and v 3 all belong to the same layer of the BFS-layering L of G 5 . Assuming that u i ∈ A x i and v i ∈ A y i , it follows that (x i , y i ) is an edge of H assigned to Q 2 and x i ≺ H y i , for i = 1, 2, 3.
By Property (P.3), either x 1 = x 2 = x 3 or y 1 = y 2 = y 3 holds. First, we prove that x 1 = x 2 = x 3 always holds. Assume for a contradiction that, without loss of generality, x 1 = x 2 . Then, y = y 1 = y 2 = y 3 holds. Since the edges of Q 2 are binding in the peeling-into-levels decomposition of H and since x 1 ≺ H x 2 ≺ H y, vertices x 1 and x 2 belong to level L j of H , while y belongs to level L j+1 , for some 0 ≤ j < λ. By Property (T.4), y has only one neighbor on level L j such that the edge connecting y to this neighbor is assigned to Q 2 . This contradicts the fact that both edges (x 1 , y) and (x 2 , y) are assigned to Q 2 , since x 1 = x 2 and x 1 and x 2 belong to L j . Hence x = x 1 = x 2 = x 3 holds. Now, since the edges of Q 2 are binding edges in the peeling-into-levels decomposition of H , and since, for i = 1, 2, 3 , x ≺ H y i , it follows that vertex x belongs to level L j of H , while vertex y i belongs to level L j+1 , for some 0 ≤ j < λ. Let τ be the tripod of bag A x and let τ 1 , τ 2 , τ 3 be the tripods of A y 1 , A y 2 and A y 3 , respectively. Note that vertices y 1 , y 2 and y 3 are not necessarily distinct, that is, the corresponding tripods τ 1 , τ 2 and τ 3 are not necessarily pairwise different. However, since u 1 ≺ G u 2 ≺ G u 3 , and since u 1 , u 2 and u 3 belong to the same layer in the BFS-layering L of G, we can conclude that u 1 , u 2 and u 3 belong to p 1 , p 2 and p 3 , respectively, where p 1 , p 2 and p 3 are the first, second and third vertical paths of τ . From the fact that, for i = 1, 2, 3, the edge (x, y i ) belongs to H , we have that τ is a parent tripod of τ i or vice versa. Since x belongs to level L j of H and vertices y 1 , y 2 and y 3 belong to level L j+1 , Lemma 4 implies that τ is a parent tripod of τ 1 , τ 2 and τ 3 . By Lemma 3, there exist at most two connected components c 1 s and c 2 s of H j+1 , such that the edges connecting x to vertices of c 1 s and c 2 s have been assigned to Q 2 . Without loss of generality, by Invariant (T.2) we assume that all vertices of c 1 s (if any) precede those of c 2 s (if any). We first argue that not all vertices y 1 , y 2 and y 3 belong to only one of c 1 s or c 2 s , which further implies that both components c 1 s and c 2 s exist. Assume to the contrary that y 1 , y 2 and y 3 belong to c 1 s ; a similar argument applies for c 2 s . By Property (P.6), tripod-vertices of component c 1 s are not adjacent to the vertices of at least one vertical path p i of τ in G, where 1 ≤ i ≤ 3. In this case, however, we obtain a contradiction to the fact that v i , which is a tripod-vertex of c 1 s , is connect to u i that belongs to p i , i.e., edge (u i , v i ) cannot exist in G.
Hence, not all vertices y 1 , y 2 and y 3 belong to only one of c 1 s or c 2 s , as we initially claimed. In particular, since y 3 H y 2 H y 1 and since all vertices of c 1 s precede those of c 2 s , it follows that y 3 belongs to c 1 s , while y 1 belongs to c 2 s . On the other hand, vertices u 1 , u 2 and u 3 belong to p 1 , p 2 and p 3 , respectively, which implies that the vertices of c 1 s and c 2 s are connected to all vertical paths of τ . Thus, Property (P.11) applies, which states that the tripod-vertices of c 2 s (including v 1 ) are not connected to the first vertical path p 1 of τ . However, this contradicts the fact that (u 1 , v 1 ) is an edge of G, with u 1 ∈ p 1 . Hence, the edges of E 2 3 cannot form a 3-rainbow. Using Property (P.12) instead of Property (P.11), we can symmetrically prove that the edges of E 3 3 cannot form a 3-rainbow. This completes the proof of the lemma. We are now ready to state the main theorem of this section by combining Lemmas 5 and 6.

Theorem 7
Every planar graph has queue number at most 42.

Conclusions
In this work, we improved the upper bound on the queue number of planar graphs by refining the three techniques involved in the original algorithm [5]. We believe that our approach has the potential to further reduce the upper bound by at least 3 more queues (i.e., from 42 to 39). However, more elaborate arguments that exploit planarity in more depth are required, and several details have to be worked out. Towards narrowing the gap with the lower bound of 4, determining the exact queue number of planar 3-trees becomes critical; an improvement of the current upper bound of 5 (to meet the lower bound of 4) implies a direct improvement on the upper bound on the queue number of general planar graphs. On the other hand, to obtain a better understanding of the general problem, it is also reasonable to further examine subclasses of planar graphs, such as bipartite planar graphs or planar graphs with bounded degree (e.g., cubic planar graphs).
Funding Open access funding provided by HEAL-Link Greece. No funding was received for conducting this study.

Declarations
Competing Interests The authors have no competing interests to declare that are relevant to the content of this article.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.