Monochromatic Clique Decompositions of Graphs

Let $G$ be a graph whose edges are coloured with $k$ colours, and $\mathcal H=(H_1,\dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $\mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1\le i\le k$. Let $\phi_{k}(n,\mathcal H)$ be the smallest number $\phi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $\mathcal H$-decomposition with at most $\phi$ elements. Extending the previous results of Liu and Sousa ["Monochromatic $K_r$-decompositions of graphs", Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in $\mathcal H$ is a clique and $n\ge n_0(\mathcal H)$ is sufficiently large.Comment: 14 pages; to appear in J Graph Theor

The optimal pebbling number of staircase graphs

Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can be placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs. © 2018 Elsevier B.V

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

The Gifted and Gifted Education in Hungary

The real challenge is to see value that is not yet in its true form. Becoming gifted is a process, during which characteristics of giftedness are present throughout, but not necessarily in a form perceptible or acceptable to the environment. Giftedness does not hide itself, only to the extent that the environment believes it hidden. Perception defines the pattern that manifests itself. The beginning of the 20th century is a success story of Hungarian gifted education. Outstanding teachers and their students have reached outstanding achievements through gifted education linked to everyday education. Their methods and ideas are durable, and are therefore worth recalling

Response of the neurovascular unit to brain metastatic breast cancer cells

Therapeutic resistance of cerebral secondary tumours largely depends on unique aspects linked to the neurovascular unit, especially cerebral endothelial cells and astrocytes. By using advanced microscopy techniques, here we explored novel mechanisms related to the neurovascular unit during extravasation and proliferation of triple negative breast cancer cells in the brain. Metastatic mammary carcinoma cells arrested and elongated within one hour in cerebral microvessels, but their number decreased by almost 80% in the first two days. Interestingly, malignant cells induced vasoconstriction and development of intraluminal endothelial plugs, which isolated invading cells from the circulation. During diapedesis - which usually took place on day four and five after inoculation of the tumour cells - continuity of cerebral endothelial tight junctions remained intact, indicating migration of cancer cells through the transcellular pathway. In addition, metastatic cells induced formation of multiluminal vessels and claudin-5-positive endothelial blebs. However, even severe endothelial blebbing could be reversed and the vessel morphology was restored shortly after the tumour cells completed transendothelial migration. Similar to neuro-inflammatory leukocytes, tumour cells migrated not only through the endothelial layer, but through the glia limitans perivascularis as well. Nevertheless, along with the growth of metastatic lesions by co-option of pre-existing capillaries, astrocytes and astrocyte end-feet were gradually expelled from the vessels to the border of the tumour. Taken together, we identified previously unknown mechanisms involved in the reaction of brain resident cells to invading breast cancer cells. Our results contribute to a better understanding of the complex cross-talk between tumour cells and host cells in the brain, which is essential for the identification of new therapeutic targets in this devastating disease