A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a d...

Given a graph G = (V, E) whose vertices have been properly coloured, we say that a path in G is colourful if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy Theorem that every properly coloured graph contains a colourful path on χ(G) vertices. It is interesting to think of what analogous result one could obtain if one considers induced colourful paths inste...

The purpose of this study is to develop a framework for understanding teachers’ knowledge for teaching mathematics (MKT) by examining a hierarchy of 317 South Korean elementary teachers’ knowledge for teaching mathematics. Through a web-based survey, the teachers were asked to rank order their beliefs about and use of knowledge subdomains for teaching mathematics. Findings show that teachers’ b...

Diwan and Mubayi asked how many edges of each color could be included in a 3-edge-colored multigraph containing no rainbow triangle. We answer this question under the modest assumption that the multigraphs in question contain at least one edge between every pair of vertices. We also conjecture that this assumption is, in fact, without loss of generality.

The problem of determining the smallest number of edges, h(n;κ ≥ r), which guarantees that any graph with n vertices and h(n;κ ≥ r) edges will contain a pair of vertices joined by r internally disjoint paths was posed by Erdös and Gallai. Bollobás considered the problem of determining the largest number of edges f(n;κ ≤ `) for graphs with n vertices and local connectivity at most `. One can see...

Given a graph G = (V,E) and a family H of graphs, a subgraph G′ of G is usually called an H-factor, if it is a spanning subgraph of G and its every component is isomorphic to some member of H. Here we focus on the case K2 ∈ H. Many nice results are known in the literature for this case. We show some very general theorems (Tutte type existence theorem, Tutte-Berge type minimax formula, Gallai-Ed...

By a result of Gallai, every finite graph G has a vertex partition into two parts each inducing an element of its cycle space. This fails for infinite graphs if, as usual, the cycle space is defined as the span of the edge sets of finite cycles in G. However we show that, for the adaptation of the cycle space to infinite graphs recently introduced by Diestel and Kühn (which involves infinite cy...

Berge’s strong path partition conjecture from 1982 generalizes and extends Dilworth’s theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture is known to be true for all digraphs only for k = 1 (by the Gallai–Milgram theorem) and for k ≥ λ (where λ is the cardinality of the longest path in the graph). The attempts made, so far, to prove the conjec...

Most head and neck squamous cell carcinoma (HNSCC) patients present with late-stage cancers, which are difficult to treat. Therefore, early diagnosis of high-risk premalignant lesions and incipient cancers is important. HNSCC is currently perceived as a single progression mechanism, resulting in immortal invasive cancers. However, we have found that approximately 40% of primary oral SCCs are mo...

We consider simple undirected graphs. An edge subset A of G is called an induced n{star packing of G if every component of the subgraph G[A] induced by A is a star with at most n edges and is an induced subgraph of G. We consider the problem of nding an induced n{star packing of G that covers the maximum number of vertices. This problem is a natural generalization of the classical matching prob...