On s-hamiltonian line graphs of claw-free graphs

For an integer $s\ge 0$, a graph $G$ is $s$-hamiltonian if for any vertex subset $S?V\left(G\right)$ with $\left|S\right|\le s$, $G?S$ is hamiltonian, and $G$ is $s$-hamiltonian connected if for any vertex subset $S?V\left(G\right)$ with $\left|S\right|\le s$, $G?S$ is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Ku?zel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjá?ek and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of $s$-hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for $s\ge 2$, a line graph $L\left(G\right)$ is $s$-hamiltonian if and only if $L\left(G\right)$ is $(s+2)$-connected. In this paper we prove the following.(i) For an integer $s\ge 2$, the line graph $L\left(G\right)$ of a claw-free graph $G$ is $s$-hamiltonian if and only if $L\left(G\right)$ is $(s+2)$-connected.(ii) The line graph $L\left(G\right)$ of a claw-free graph $G$ is 1-hamiltonian connected if and only if $L\left(G\right)$ is 4-connected.     

Synthesis and radical polymerization of methacrylate endowed with bicyclobis(γ‐butyrolactone) moiety through methylene linker

A series of methacrylates bearing bicyclobis(γ‐butyrolactone) (BBL) moiety were synthesized and radically polymerized to afford the corresponding poly(methacrylate)s bearing BBL moiety in the side chain, with expecting that the high polarity and rigidity of BBL would be inherited by the polymers. The resulting polymers were soluble in polar aprotic solvents such as dimethyl sulfoxide and N,N‐dimethylformamide because of the high polarity of the BBL moiety. The glass transition temperatures (T_g) of the polymers depended on the length of methylene linker that tethered the methacrylate and BBL moieties, making the use of shorter linkers lead to higher T_gs. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 2462–2468     

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

We show that for every there exists C > 0 such that if then asymptotically almost surely the random graph contains the k^th power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi‐polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi‐polynomial algorithm for finding a tight Hamilton cycle in the random k‐uniform hypergraph for . The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of p. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.     

Uniform random colored complexes

We present here random distributions on (D + 1)‐edge‐colored, bipartite graphs with a fixed number of vertices 2p. These graphs encode D‐dimensional orientable colored complexes. We investigate the behavior of those graphs as p→∞. The techniques involved in this study also yield a Central Limit Theorem for the genus of a uniform map of order p, as p→∞.     

A new arithmetic criterion for graphs being determined by their generalized Q-spectrum

“Which graphs are determined by their spectrum (DS for short)?” is a fundamental question in spectral graph theory. It is generally very hard to show a given graph to be DS and few results about DS graphs are known in literature. In this paper, we consider the above problem in the context of the generalized $Q$-spectrum. A graph $G$ is said to be determined by the generalized $Q$-spectrum (DGQS for short) if, for any graph $H$, $H$ and $G$ have the same $Q$-spectrum and so do their complements imply that $H$ is isomorphic to $G$. We give a simple arithmetic condition for a graph being DGQS. More precisely, let $G$ be a graph with adjacency matrix $A$ and degree diagonal matrix $D$. Let $Q=A+D$ be the signless Laplacian matrix of $G$, and $W_Q\left(G\right)=[e,Qe,\dots ,Q^{n-1}e]$ ($e$ is the all-ones vector) be the $Q$-walk matrix. We show that if $\frac{det{W}_Q\left(G\right)}{2^{\lfloor \frac{3n-2}{2}\rfloor }}$ (which is always an integer) is odd and square-free, then $G$ is DGQS.     

Meyniel's conjecture holds for random d‐regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3.