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...

A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic toK1,3. The kth power of a graphG, denoted byGk , is the graph with vertex set V (G) in which two vertices are adjacent if and only if the distance between them in G is at most k. I...

In 1968, Erdös and Lovász conjectured that for every graph G and all integers s, t ≥ 2 such that s + t − 1 = χ(G) > ω(G), there exists a partition (S, T ) of the vertex set of G such that χ(G|S) ≥ s and χ(G|T ) ≥ t. For general graphs, the only settled cases of the conjecture are when s and t are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stabili...

An efficiently total dominating set of a graph G is a subset of its vertices such that each vertex of G is adjacent to exactly one vertex of the subset. If there is such a subset, then G is an efficiently total dominable graph (G is etd). In this paper, we prove NP-completeness of the etd decision problem on the class of planar bipartite graphs of maximum degree 3. Furthermore, we give an effic...

A graph G is locally connected if the subgraph induced by the neighbourhood of each vertex is connected. We prove that a locally connected graph G of order p 3, containing no induced subgraph isomorphic to K 1;3 , is Hamilton-connected if and only if G is 3-connected.