Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length l for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph on n > 4k^4 vertices with independence number k, then G is pancyclic. He then suggested that n = \Omega(k^2) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n > ck^{7/3} suffices

Long paths and cycles in random subgraphs of graphs with large minimum degree

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$ that tends to infinity as $k$ does, then $G_p$ asymptotically almost surely contains a cycle (and thus a path) of length at least $(1-o(1))k$, and (ii) if $p \ge (1+o(1))\ln k/k$, then $G_p$ asymptotically almost surely contains a path of length at least $k$. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking $G$ to be the complete graph on $k+1$ vertices.Comment: 26 page

Bandwidth theorem for random graphs

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every positive $r,\Delta,\gamma$, there exists $\beta$ such that if $H$ is an $n$-vertex $r$-chromatic graph with maximum degree at most $\Delta$ which has bandwidth at most $\beta n$, then any graph $G$ on $n$ vertices with minimum degree at least $(1 - 1/r + \gamma)n$ contains a copy of $H$ for large enough $n$. In this paper, we extend this theorem to dense random graphs. For bipartite $H$, this answers an open question of B\"{o}ttcher, Kohayakawa, and Taraz. It appears that for non-bipartite $H$ the direct extension is not possible, and one needs in addition that some vertices of $H$ have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed $r$-chromatic graph $H_0$ which one can find in a spanning subgraph of $G(n,p)$ with minimum degree $(1-1/r + \gamma)np$.Comment: 29 pages, 3 figure

Ramsey numbers of cubes versus cliques

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.Comment: 26 page

Processing Unfamiliar Words: Strategies, Knowledge Sources, and the Relationship to Text and Word Comprehension

This study examines strategies (inferencing and ignoring) and knowledge sources (semantics, morphology, paralinguistics, etc.) that second language learners of English use to process unfamiliar words in listening comprehension and whether the use of strategies or knowledge sources relates to successful text comprehension or word comprehension. Data were collected using the procedures of immediate retrospection without recall support and of stimulated recall. Twenty participants with Chinese as their first language participated in the procedures. Both qualitative and quantitative analyses were made. The results indicate that inferencing is the primary strategy that learners use to process unfamiliar words in listening and that it relates to successful text comprehension. Among the different knowledge sources that learners use, the most frequently used knowledge sources are semantic knowledge of words in the local co-text combined with background knowledge and semantic knowledge of the overall co-text. The finding that the use of most knowledge sources does not relate to the comprehension of the word suggests that no particular knowledge source is universally effective or ineffective and that what is crucial is to use the various knowledge sources flexibly. Résumé Cette étude examine les stratégies (la déduction et l'omission de mots) et les sources de connaissances (sémantique, morphologie, connaissance antérieure, etc.) utilisées par les étudiants d’anglais langue seconde (ALS) pour comprendre les mots inconnus à l'oral, et s'interroge sur les liens entre l’emploi des stratégies ou sources de connaissances et la bonne compréhension des textes et des mots. Les données ont été recueillies immédiatement après observation, sans rappel ni simulation ultérieure. Vingt locuteurs de langue maternelle chinoise ont participé à l’étude. Des approches qualitative et quantitative ont été utilisées. Les résultats indiquent que la déduction est la stratégie de toute premiѐre importance utilisée par les sujets pour comprendre les mots inconnus à l'oral, et ceci est lié à une bonne compréhension du texte. Parmi les sources de connaissances, celles qui sont les plus souvent utilisées par les étudiants sont la connaissance sémantique des mots du contexte immédiat alliée avec la connaissance de fond et la connaissance sémantique du texte global. Les résultats indiquent que l'emploi de la plupart des sources de connaissances n’a aucun rapport avec la compréhension des mots, suggérant ainsi qu' aucune source de connaissance en particulier n'est universellement efficace ou inefficace . Ce qui est crucial est l’emploi flexible de diverses sources de connaissances.