Two-Page Book Embeddings of 4-Planar Graphs

Back in the Eighties, Heath 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.Comment: 21 pages, 16 Figures. A shorter version is to appear at STACS 201

On the tractability of some natural packing, covering and partitioning problems

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mathrm{B}$? \textbf{Covering problem:} can we cover $E$ with an object of type $\mathrm{A}$, and an object of type $\mathrm{B}$? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an $s$-$t$ path $P$ and an $s'$-$t'$ path $P'$ that are edge-disjoint. However, many others were not, for example can we find an $s$-$t$ path $P\subseteq E$ and a spanning tree $T\subseteq E$ that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

AbstractIn 1956, W.T. Tutte proved that every 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. It is shown that Sanders’ result is best possible by constructing 4-connected maximal planar graphs with three edges a large distance apart such that any hamiltonian cycle misses one of them. If the maximal planar graph is 5-connected then such a construction is impossible