Conflict-free coloring of graphs

We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page

On covering expander graphs by Hamilton cycles

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree $\Delta$ satisfies some basic expansion properties and contains a family of $(1-o(1))\Delta/2$ edge disjoint Hamilton cycles, then there also exists a covering of its edges by $(1+o(1))\Delta/2$ Hamilton cycles. This implies that for every $\alpha >0$ and every $p \geq n^{\alpha-1}$ there exists a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some math/061275

GRANTS, PAYMENTS, LAWS - REVIEW AND SUGGESTION FOR STUDENT FINANCING MODELS

Topic of the student financing is somehow similar to the one of the football in Hungary; almost everyone has an opinion about it. It is mostly the introduction and abrogation of the tuition fees since when the public opinion has been paying particular attention to the changes. Looking back to the past almost twenty years, the author of the article keeps his eyes on the changes of the national student allowance-fee system. Following a historical-like introduction, he puts forward a proposal to a new model concerning basically the student subsidies not to be refund. According to the new system, the study and social subsidies will help a broader circle - on the other hand a less one than the current proportion - of students, abandoning at the same time the multi-channel, frittered away, non-system approached distribution of the central governmental sources, and the indirect subsidies (for the institutions)