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

Valóság és realizmus Dante főművében = Realtà e realismo nell'opera principale di Dante

Reality and Realism in Dante’s Masterpiece In the paper we would like to present some aspects of the Divine Comedy concerning his mode of description. We are aware of the importance of the religious factors going through the entire work but we emphasize in this paper his inclination to observe the social, political and natural aspects of his age. According to some specialists of Dante works like Péterfy, Kasztner, Auerbach and others (like for example also Pál) we point out mainly the social and natural comparison reaveling the exceptional Dante’s capacity to observe the characteristic features of Italian landscape describing shepherds, flocks, mountains, waterfalls, flowers etc. This inclination towards natural phenomenons is in harmony with the other artist of his age: the famous painting (friend of Dante) Giotto di Bondone who in Padova has created a new type of representation of the reality. The comparison of the two artists show some not accidental coincidances contemplating the world around