Back in the eighties, Heath [6] showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book embedding viewpoint of the problem.Lemma 2. Given a 4-planar triconnected graph G and a separating trianglepairwise distinct or all represent the same vertex. Proof. In the other case, where w.l.o.g. A in = B in = v and Γ in = v, there exists a separation pair (v, Γ) contradicting the triconnectivity of G. A symmetric argument applies to A out , B out , Γ out . Lemma 3. In a 4-planar triconnected graph, every pair of distinct separating triangles T and T is vertex disjoint, i.e. V (T ) ∩ V (T ) = ∅.Proof. Assume to the contrary that T and T share an edge or a vertex. In the first case, let w.l.o.g. e = (u, v) be the common edge. The degree of both u and v is at least five, since three edges are required for T , T and two additional edges to connect G in (T ) and G in (T ) to T and T , respectively. In the second case, let v denote the common vertex. Since v is part of two edge disjoint cycles and connected to G in (T ) and G in (T ), it follows that deg(v) ≥ 6.Consider now a 4-planar triconnected graph with a single separating triangle T . Similar to Chen [2], the idea is to compute two cycles H in (T ) and H out (T ) for G in (T ) and G out (T ) and link them via the separating triangle together. The crucial observation is that if two cycles intersect as illustrated in Fig. 2, i.e., they contain two edges of the triangle but have only one of them in common, then we can always merge them into one cycle. Lemma 4. Let G be a triconnected 4-planar graph, T a separating triangle, and H in (T ) and H out (T ) two subhamiltonian cycles for G in (T ) and G out (T ), resp. If E(H in (T )) ∩ E(T ) = {e in , e} and E(H out (T )) ∩ E(T ) = {e out , e} where {e, e in , e out } are the edges of T , then G is subhamiltonian. Proof. Let w.l.o.g. e = (A, B), e in = (B, Γ) and e out = (A, Γ) as illustrated in Fig. 2. The result of removing the edges of T from both cycles are two paths P out = B Γ and P in = Γ A. Joining them at Γ and inserting e yields a subhamiltonian cycle.

In a book embedding, the vertices of a graph are placed on the "spine" of a book and the edges are assigned to "pages", so that edges on the same page do not cross. In this paper, we prove that every 1-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is O( √ n), where n is the number of vertices of the graph. IntroductionA book embedding is a special type of a graph embedding, in which (i) the vertices of the graph are restricted to a line along the spine of a book, and, (ii) the edges on the pages of the book in such a way that edges residing on the same page do not cross. The minimum number of pages required to construct such an embedding is known as book thickness or page number of a graph. An obvious upper bound on the page number of an n-vertex graph is n/2 , which is tight for complete graphs [3]. Book embeddings have a long history of research dating back to early seventies [19]. Therefore, there is a rich body of literature (see, e.g., [4] and [20]). For the class of planar graphs, a central result is due to Yannakakis [23], who in the late eighties proved that planar graphs have book thickness at most four. It remains, however, unanswered whether the known bound of four is tight. Heath [10], for example, proves that all planar 3-trees are 3-page book embeddable. For more restricted subclasses of planar graphs, Bernhart and Kainen [3] show that the graphs with book thickness one are the outerplanar graphs, while the class of two-page embeddable graphs coincides with the class of subhamiltonian graphs (recall that subhamiltonian is a graph that is a subgraph of a planar Hamiltonian graph). Testing whether a graph is subhamiltonian is NP-complete [22]. However, several graph classes are known to be subhamiltonian (and therefore two-page book embeddable), e.g., 4-connected planar graphs [18], planar graphs without separating triangles [13], Halin graphs [7], planar graphs with maximum degree 3 or 4 [11,2].In this paper, we go a step beyond planar graphs. In particular, we consider 1-planar graphs and prove that their book thickness is constant. Recall that a graph is 1-planar, if it admits a drawing in which each edge is crossed at most once. To the best of our knowledge, the only (nontrivial) upper bound on the book thickness of 1-planar graphs on n vertices is O( √ n). This is * Electronic address: bekos@informatik. In the remainder of this paper, we will assume that a simple 1-planar drawing Γ(G) of the input 1-planar graph G is also specified as part of the input of the problem. This is due to a result of Grigoriev and Bodlaender [9], and, independently of Kohrzik and Mohar [14], who proved that the problem of determining whether a graph is 1-planar is NP-hard (note that the problem remains NP-hard, even if the deletion of a single edge makes the graph planar [6]). In addition, we assume biconectivity, as it...

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Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, diameter, and acyclicity of the monochromatic components. We focus on defective colorings with $\kappa$ colors in which the monochromatic components are acyclic and have small diameter, namely we consider (edge, $\kappa$)-colorings, in which the monochromatic components have diameter $1$, and (star, $\kappa$)-colorings, in which they have diameter $2$. We prove that the (edge, $3$)-coloring problem remains NP-complete even for graphs with maximum vertex-degree $6$, hence answering an open question posed by Cowen et al. [Cowen, Goddard, Jesurum, J. Graph Theory, 1997], and for planar graphs with maximum vertex-degree $7$, and we prove that the (star, $3$)-coloring problem is NP-complete even for planar graphs with bounded maximum vertex-degree. On the other hand, we give linear-time algorithms for testing the existence of (edge, $2$)-colorings and of (star, $2$)-colorings of partial $2$-trees. Finally, we prove that outerpaths, a notable subclass of outerplanar graphs, always admit (star, $2$)-colorings.

In a book embedding, the vertices of a graph are placed on the "spine" of a book and the edges are assigned to "pages", so that edges on the same page do not cross. In this paper, we prove that every 1-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is O( √ n), where n is the number of vertices of the graph.