Zdenek Ryjácek

Claw-free graphs—a survey

Abstract In this paper we summarize known results on claw-free graphs. The paper is subdivided into the following chapters and sections: 1. 1. Introduction 2. 2. Paths, cycles, hamiltonicity 2.1. (a) Preliminaries 2.2. (b) Degree and neighborhood conditions 2.3. (c) Local connectivity conditions 2.4. (d) Further forbidden subgraphs 2.5. (e) Invariants 2.6. (f) Squares 2.7. (g) Regular graphs 2.8. (h) Other hamiltonicity related results and generalizations 2.9. 3. Matchings and factors 2.10. 4. Independence, domination, other invariants and extremal problems 2.11. 5. Algorithmic aspects 2.12. 6. Miscellaneous 2.13. 7. Appendix — List of all 2-connected nonhamiltonian claw-free graphs on n ⩽ 12 vertices.

Closure, 2-factors, and cycle coverings in claw-free graphs

A main result proved in this paper is the following. Theorem. Let G be a noncomplete graph on n vertices with degree sequence d1 ≥ d2 ≥ · · · ≥ dn and t ≥ 2 be a prime. Let m = gcd{t, di - dj: 1 ≤ i < j ≤ n} and set

Closure and Hamiltonian-connectivity of claw-free graphs

d =\cases{1\ \ \ if\ m = t\ and\ \ m \not\mid\ d_{i}\ for\ 1 \leq i \leq n \cr 0\ \ \ otherwise.}

Forbidden subgraphs that imply 2-factors

Then R(tG, ℤt) = t(n + d) - d, where R is the zero-sum Ramsey number. This settles, almost completely, problems raised in [Bialostocki & Dierker, J Graph Theory, 1994; Y. Caro, J Graph Theory, 1991].

Strengthening the closure concept in claw-free graphs

Let G be a graph on n 3 vertices. Diracs minimum degree condition is the condition that all vertices of G have degree at least . This is a well-known sufficient condition for the existence of a Hamilton cycle in G. We give related sufficiency conditions for the existence of a Hamilton cycle or a perfect matching involving a restriction of Diracs minimum degree condition to certain subsets of the vertices. For this purpose we define G to be 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of G isomorphic to K1,3 (a claw) has (have) degree at least . Thus, every claw-free graph is 2-heavy, and every 2-heavy graph is 1-heavy. We show that a 1-heavy or a 2-heavy graph G has a Hamilton cycle or a perfect matching if we impose certain additional conditions on G involving numbers of common neighbours, (local) connectivity, and forbidden induced subgraphs. These results generalize or extend previous work of Broersma & Veldman, Dirac, Fan, Faudree et al., Goodman & Hedetniemi, Las Vergnas, Oberly & Sumner, Ore, Shi, and Sumner.

How Many Conjectures Can You Stand? A Survey

Abstract In Ryjacek (1997), the closure cl( G ) for a claw-free graph G is defined, and it is proved that G is hamiltonian if and only if cl( G ) is hamiltonian. On the other hand, there exist infinitely many claw-free graphs G such that G is not hamiltonian-connected (resp. homogeneously traceable) while cl( G ) is hamiltonian-connected (resp. homogeneously traceable). In this paper we define a new closure cl k ( G ) ( k ⩾ 1) as a generalization of cl( G ) and prove the following theorems. (1) A claw-free graph G is hamiltonian-connected if and only if cl 3 ( G ) is hamiltonian-connected. (2) A claw-free graph G is homogeneously traceable if and only if cl 2 ( G ) is homogeneously traceable. We also discuss the uniqueness of the closure.

A note on degree conditions for Hamiltonicity in 2-connected claw-free graphs

There are many results concerned with the hamiltonicity of K1,3-free graphs. In the paper we show that one of the sufficient conditions for the K1,3-free graph to be Hamiltonian can be improved using the concept of second-type vertex neighborhood. The paper is concluded with a conjecture.

Factor-criticality and matching extension in DCT-graphs

The connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph is hamiltonian have been characterized by Bedrossian [Forbidden subgraph and minimum degree conditions for hamiltonicity, Ph.D. Thesis, Memphis State University, 1991], and extensions of these excluding graphs for general graphs of order at least 10 were proved by Faudree and Gould [Characterizing forbidden pairs for Hamiltonian properties, Discrete Math. 173 (1997) 45-60]. In this paper a complete characterization of connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph of order at least 10 has a 2-factor will be proved. In particular it will be shown that the characterization for 2-factors is very similar to that for hamiltonian cycles, except there are seven additional pairs. In the case of graphs of all possible orders, there are four additional forbidden pairs not in the hamiltonian characterization, but a claw is part of each pair.