On planar hypohamiltonian graphs

We present a planar hypohamiltonian graph on 42 vertices and (as a corollary) a planar hypotraceable graph on 162 vertices, improving the bounds of Zamfirescu and Zamfirescu and show some other consequences. We also settle the open problem whether there exists a positive integer N, such that for every integer n≥N there exists a planar hypohamiltonian/hypotraceable graph on n vertices. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 55‐68, 2011     

On paths in planar graphs

This paper generalizes a theorem of Thomassen on paths in planar graphs. As a corollary, it is shown that every 4-connected planar graph has a Hamilton path between any two specified vertices x, y and containing any specified edge other than xy. © 1997 John Wiley & Sons, Inc.     

On acyclic colorings of planar graphs

The conjecture of B. Grünbaum on existing of admissible vertex coloring of every planar graph with 5 colors, in which every bichromatic subgraph is acyclic, is proved and some corollaries of this result are discussed in the present paper.     

On stable cutsets in claw-free graphs and planar graphs

A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let $K_4$ and $K_{1,3}$ (claw) denote the complete (bipartite) graph on 4 and $1+3$ vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a $K_4$-free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free graph with maximum degree at most four or whether a (claw, $K_4$)-free graph admits a stable cutset. As a by-product we obtain that the stable cutset problem is polynomially solvable for claw-free planar graphs, and also for planar line graphs.Thus, the computational complexity of the stable cutset problem is completely determined for claw-free graphs with respect to degree constraint, and for claw-free planar graphs. Moreover, we prove that the stable cutset problem remains NP-complete for $K_4$-free planar graphs with maximum degree five.     

On acyclic colorings of planar graphs

The conjecture of B. Grünbaum on existing of admissible vertex coloring of every planar graph with 5 colors, in which every bichromatic subgraph is acyclic, is proved and some corollaries of this result are discussed in the present paper.     

Smaller planar triangle-free graphs that are not 3-list-colorable

In 1995, Voigt constructed a planar triangle-free graph that is not 3-list-colorable. It has 166 vertices. Gutner then constructed such a graph with 164 vertices. We present two more graphs with these properties. The first graph has 97 vertices and a failing list assignment using triples from a set of six colors, while the second has 109 vertices and a failing list assignment using triples from a set of five colors.     

On shortest disjoint paths in planar graphs

For a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}, the k disjoint paths problem is to find k vertex-disjoint paths P₁,…,P_k, where P_i is a path from s_i to t_i for each i=1,…,k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P_i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k=2,3.Min-Sum: We extend recent results by Colin de Verdière and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time.Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3.     

On certain Hamiltonian cycles in planar graphs

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least five. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than five contains at least distinct Hamiltonian cycles. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 81–96, 1999     

On the delta-wye reduction for planar graphs

We provide an elementary proof of an important theorem by G. V. Epifanov, according to which every two-terminal planar graph satisfying certain connectivity restrictions can by some sequence of series/parallel reductions and delta-wye exchanges be reduced to the graph consisting of the two terminals and just one edge.     

On the connectivity of maximal planar graphs

Let d₁ ? d₂ ? … ? d_p be the vertex degrees of a maximal planar graph G. Etourneau has shown that if d₁ ? 6 and d_p = 5, then G is 5-connected. We generalize Etourneau's result by giving sufficient conditions in terms of the vertex degrees for G to be d_p -connected.     

On 2-walks in chordal planar graphs

A 2-walk is a closed spanning trail which uses every vertex at most twice. A graph is said to be chordal if each cycle different from a 3-cycle has a chord. We prove that every chordal planar graph G with toughness has a 2-walk.     

On Hamilton cycles in certain planar graphs

Let G be a 2-connected plane graph with outer cycle X_G such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of X_G. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.     

On the toughness index of planar graphs

The toughness indexτ(G) of a graph G is defined to be the largest integer t such that for any S ? V(G) with |S| > t, c(G - S) < |S| - t, where c(G - S) denotes the number of components of G - S. In particular, 1-tough graphs are exactly those graphs for which τ(G) ≥ 0. In this paper, it is shown that if G is a planar graph, then τ(G) ≥ 2 if and only if G is 4-connected. This result suggests that there may be a polynomial-time algorithm for determining whether a planar graph is 1-tough, even though the problem for general graphs is NP-hard. The result can be restated as follows: a planar graph is 4-connected if and only if it remains 1-tough whenever two vertices are removed. Hence it establishes a weakened version of a conjecture, due to M. D. Plummer, that removing 2 vertices from a 4-connected planar graph yields a Hamiltonian graph.     

On circuit decomposition of planar Eulerian graphs

We give a common generalization of P. Seymour's “Integer sum of circuits” theorem and the first author's theorem on decomposition of planar Eulerian graphs into circuits without forbidden transitions.     

On planar intersection graphs with forbidden subgraphs

Let be a family of n compact connected sets in the plane, whose intersection graph has no complete bipartite subgraph with k vertices in each of its classes. Then has at most n times a polylogarithmic number of edges, where the exponent of the logarithmic factor depends on k. In the case where consists of convex sets, we improve this bound to O(n log n). If in addition k = 2, the bound can be further improved to O(n). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 205–214, 2008     

On cuts and matchings in planar graphs

We study the max cut problem in graphs not contractible toK ₅, and optimum perfect matchings in planar graphs. We prove that both problems can be formulated as polynomial size linear programs.Supported by the joint project Combinatorial Optimization of the Natural Sciences and Engineering Research Council of Canada and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

On the linear arboricity of planar graphs

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. Akiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ. The conjecture has been proved to be true for graphs having Δ = 1, 2, 3, 4, 5, 6, 8, 10. Combining these results, we prove in the article that the conjecture is true for planar graphs having Δ(G) ≠ 7. Several related results assuming some conditions on the girth are obtained as well. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 129–134, 1999     

On the tightness conditions for maximal planar graphs

We characterize the tight structure of a vertex-accumulation-free maximal planar graph with no separating triangles. Together with the result of Halin who gave an equivalent form for such graphs, this yields that a tight structure always exists in every 4-connected maximal planar graph with one end.