A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O (

Let G be a graph of order n. De.ne fk(G) (Fk(G)) to be the minimum (maximum) number of components in a k-factor of G. For convenience, we will say that fk(G)=0 if G does not contain a k-factor. It is known that if G is a claw-free graph with su4ciently high minimum degree and proper order parity, then G contains a k-factor. In this paper we show that f2(G)6n= for n and su4ciently large and G cl...

A graph G is triangularly connected if for every pair of edges e1, e2 ∈ E(G), G has a sequence of 3-cycles C1, C2, · · · , Cl such that e1 ∈ C1, e2 ∈ Cl and such that E(Ci) ∩ E(Ci+1) 6= ∅, (1 ≤ i ≤ l − 1). In this paper it is shown that every triangularly connected claw-free graph G with |E(G)| ≥ 3 is vertex pancyclic. This implies the former results in [2], [3], [4] and [5] that every connecte...

A fast algorithm to remove proper and homogenous pairs of cliques (while preserving some graph invariants) Abstract We introduce a family of reductions for removing proper and homogeneous pairs of cliques from a graph G. This family generalizes some routines presented in the literature, mostly in the context of claw-free graphs. These reductions can be embedded in a simple algorithm that in at ...

In this paper, we show that if G is an l-connected claw-free graph with minimum degree at least three and l ∈ {2, 3}, then for any maximum independent set S, there exists a 2-factor in which each cycle contains at least l − 1 vertices in S.

Let G be a connected claw-free graph on n vertices and G be its complement. Let μ(G) be the spectral radius of G. Denote by Nn−3,3 the graph consisting of Kn−3 and three disjoint pendent edges. In this note we prove that: (1) If μ(G) ≥ n − 4, then G is traceable unless G = Nn−3,3. (2) If μ(G) ≤ μ(Nn−3,3) and n ≥ 24, then G is traceable unless G = Nn−3,3. Our works are counterparts on claw-free ...

A lemma of Fouquet implies that a claw-free graph contains an induced C5, contains no odd hole, or is quasi-line. In this paper we use this result to give an improved shortest-oddhole algorithm for claw-free graphs by exploiting the structural relationship between line graphs and quasi-line graphs suggested by Chudnovsky and Seymour’s structure theorem for quasi-line graphs. Our approach involv...

In decapod crustaceans, claw muscle undergoes atrophy in response to elevated ecdysteroids while thoracic muscle undergoes atrophy in response to unweighting. The signaling pathways that regulate muscle atrophy in crustaceans are largely unknown. Myostatin is a negative regulator of muscle growth in mammals, and a myostatin-like cDNA is preferentially expressed in muscle of the land crab, Gecar...

Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs Jinquan Xu, Ping Li, Zhengke Miao, Keke Wang & Hong-Jian Lai a Department of Mathematics, HuiZhou University, HuiZhou, Guangdong 561007, China b Department of Mathematics, Beijing Jiaotong University, Beijing, China c School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 2211...

Let G be a connected, locally connected, claw-free graph of order n and x, y be two vertices of G. In this paper, we prove that if for any 2-cut S of G, S ∩ {x, y} = ∅, then each (x, y)-path of length less than n − 1 in G is extendable, that is, for any path P joining x and y of length h(<n− 1), there exists a path P ′ in G joining x and y such that V (P ) ⊂ V (P ′) and |P ′| = h+ 1. This gener...