For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K1,4-free graph G with minimum degree δ(G) ≥ k+ 3 is pancyclic mod k and every claw-free graph G with δ(G) ≥ k+ 1 is pancyclic mod k, which confirms Thomassen’s conjecture [8] for claw-free graphs.

If G is a claw-free graph, then there is a graph cl(G) such that (i) G is a spanning subgraph of cl(G), (ii) cl(G) is a line graph of a triangle-free graph, and (iii) the length of a longest cycle in G and in cl(G) is the same. A suucient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs and the hamiltonicity of 7-connected c...

If C is a subclass of the class of claw-free graphs, then C is said to be stable if, for any G ∈ C, the local completion of G at any vertex is also in C. If cl is a closure operation that turns a claw-free graph into a line graph by a series of local completions and C is stable, then cl(G) ∈ C for any G ∈ C. In the talk we consider stability of hereditary classes of claw-free graphs defined in ...

We show a construction that gives an infinite family of claw-free graphs of connectivity κ = 2, 3, 4, 5 with complete closure and without a cycle of a given fixed length. This construction disproves a conjecture by the first author, A. Saito and R.H. Schelp.

A Λ-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G. A graph is clawfree if it does not have a subgraph isomorphic to K1,3. Our results include the following. Let G be a 3-connected claw-free graph, x ∈ V (G), e = xy ∈ E(G), and L a 3-vertex path in G. Then (c1) if v(G) = 0 mod 3, then G has a Λ-factor containing ...

Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In this paper, we provide a polynomial-time algorithm for recognizing such graphs, and more importantly for building a suitable representation.

A graph is prismatic if for every triangle T , every vertex not in T has exactly one neighbour in T . In this paper and the next in this series, we prove a structure theorem describing all prismatic graphs. This breaks into two cases depending whether the graph is 3-colourable or not, and in this paper we handle the 3-colourable case. (Indeed we handle a slight generalization of being 3-coloura...

Digital end organs composed of hard, modified epidermis, generally referred to as claws, are present in mammals and reptiles as well as in several non-amniote taxa such as clawed salamanders and frogs, including Xenopus laevis. So far, only the claws and nails of mammals have been characterized extensively and the question of whether claws were present in the common ancestor of all extant tetra...

Grooming claws are present on the second pedal digits of strepsirhines and on the second and third pedal digits of tarsiers. However, their presence in New World monkeys is often overlooked. As such, the absence of a grooming claw is generally considered an anthropoid synapomorphy. This study utilizes a quantitative multivariate analysis to define grooming claw morphology and document its prese...

Stable isotopes derived from the claws of birds could be used to determine the migratory origins of birds if the time periods represented in excised sections of claws were known. We investigated new keratin growth in the claws of adult female Lesser Scaup (Aythya affinis) by estimating the equilibration rates of stable isotopes (δ (13)C, δ (15)N, and δ (2)H) from the breeding grounds into 1 mm ...