Evelyne Flandrin

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.

Hamiltonism, degree sum and neighborhood intersections

Abstract We give a sufficient condition for hamiltonism of a 2-connected graph involving the degree sum and the neighborhood intersection of any three independent vertices.

Some properties of 3-domination-critical graphs

Abstract A graph G is 3- γ -critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. Wojcicka conjectured that every 3- γ -critical graph with minimum degree δ ⩾2 has a hamiltonian cycle. In this paper, we prove that if G is a 3- γ -critical connected graph of order n with minimum degree δ ⩾2, then (1) G is 1-tough; (2) the circumference of G is at least n −1.

Mengerian properties, hamiltonicity, and claw-free graphs

In relation to the Property Pd,m, we define two parameters generalizing the diameter and the independence number of a graph, study their behaviors, and find new conditions depending on those parameters for a claw-free graph to be Hamiltonian.

ON 2-FACTORS IN CLAW-FREE GRAPHS

Abstract A graph is said claw-free if it contains no induced subgraph isomorphic to K1,3. We prove that if G is a claw-free graph with minimum degree δ⩾4, then G contains a 2-factor with at most 6n/(δ+2)−1 components. Moreover, together with a theorem of Choudoum and Paulraj (J. Graph Theory 15 (1991) 259–265) and one of Anstee (J. Algorithms 6 (1985) 112–131), it is polynomial (in O(n3)) to construct such a 2-factor.

Neighborhood unions and extremal spanning trees

We generalize a known sufficient condition for the traceability of a graph to a condition for the existence of a spanning tree with a bounded number of leaves. Both of the conditions involve neighborhood unions. Further, we present two results on spanning spiders (trees with a single branching vertex). We pose a number of open questions concerning extremal spanning trees.

Pancyclism in hamiltonian graphs

We prove the following theorem. If G is a hamiltonian, nonbipartite graph of minimum degree at least (2n+1)5, where n represents the order of G, then G is pancyclic.

Sequences, claws and cyclability of graphs

A subset S of vertices of a graph G is called cyclable in G if there is in G some cycle containing all the vertices of S. We give two results on the cyclability of a vertex subset in graphs, one of which is related to “hamiltonian-nice-sequence” conditions and the other of which is related to “claw-free” conditions. They imply many known results on hamiltonian graph theory. Moreover, the analogous results related to the hamilton-connectivity or to the existence of dominating cycle are also given.

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

Abstract By using the closure concept introduced by the last author, we prove that for any sufficiently large nonhamiltonian claw-free graph G satisfying a degree condition of type σk(G)>n+k2−4k+7 (where k is a constant), the closure of G can be covered by at most k−1 cliques. Using structural properties of the closure concept, we show a method for characterizing all such nonhamiltonian exceptional graphs with limited clique covering number. The method is explicitly carried out for k⩽6 and illustrated by proving that every 2-connected claw-free graph G of order n⩾77 with δ(G)⩾14 and σ6(G)>n+19 is either hamiltonian or belongs to a family of easily described exceptions.

Pancyclism and small cycles in graphs

We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u) + d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u) + d(v) ≥ n + z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m for each 3 ≤ m ≤ max (dC(u, v) + 1, n+19 13 ), dC(u, v) being the distance of u and v on a hamiltonian cycle of G.