Zdeněk Ryjáček

Closure and stable Hamiltonian properties in claw-free graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes.

Forbidden subgraphs, hamiltonicity and closure in claw-free graphs

Abstract We study the stability of some classes of graphs defined in terms of forbidden subgraphs under the closure operation introduced by the second author. Using these results, we prove that every 2-connected claw-free and P7-free, or claw-free and Z4-free, or claw-free and eiffel-free graph is either hamiltonian or belongs to a certain class of exceptions (all of them having connectivity 2).

Almost claw-free graphs

We say that G is almost claw-free if the vertices that are centers of induced claws (K1,3) in G are independent and their neighborhoods are 2-dominated. Clearly, every claw-free graph is almost claw-free. It is shown that (i) every even connected almost claw-free graph has a perfect matching and (ii) every nontrivial locally connected K1,4-free almost claw-free graph is fully cycle extendable.

Closure concepts - a survey

Abstract. In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C(u,v) be a condition on two nonadjacent vertices u and v of a graph G. Then G+uv has property P if and only if G has property P. The first and now well-known result of this type was established by Bondy and Chvátal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G+uv is hamiltonian if and only if G is hamiltonian. Based on this result, they defined the n-closure cln (G) of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cln(G) is well-defined, and that G is hamiltonian if and only if cln(G) is hamiltonian. Moreover, they showed that cln(G) can be obtained by a polynomial algorithm, and that a Hamilton cycle in cln(G) can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cln(G)=Kn (and n≥3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic sufficient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their first paper on closures, Bondy and Chvátal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their first results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years.

The Prism Over the Middle-levels Graph is Hamiltonian

Let Bk be the bipartite graph defined by the subsets of {1,…,2k + 1} of size k and k + 1. We prove that the prism over Bk is hamiltonian. We also show that Bk has a closed spanning 2-trail.

Pairs of Heavy Subgraphs for Hamiltonicity of 2-Connected Graphs

Let

Forbidden subgraphs, stability and hamiltonicity

G

Clique covering and degree conditions for hamiltonicity in claw-free graphs

be a graph on

The rainbow connection number of 2-connected graphs

n

Hamiltonian index is NP-complete

vertices. An induced subgraph