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Finding a large hidden clique in a random graph
This paper presents an efficient algorithm for finding a hidden clique of vertices of size k that is based on the spectral properties of the graph and improves the trivial case k ) cn log n .
Pseudo-random Graphs
Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake,
Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions
It is proved that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H.
Coloring Graphs with Sparse Neighborhoods
It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d2/f is at most
The Largest Eigenvalue of Sparse Random Graphs
We prove that, for all values of the edge probability $p(n)$, the largest eigenvalue of the random graph $G(n, p)$ satisfies almost surely $\lambda_1(G)=(1+o(1))\max\{\sqrt{\Delta}, np\}$, where Δ is
The Number of Edge Colorings with no Monochromatic Cliques
Let F(n,r,k) denote the maximum possible number of distinct edge‐colorings of a simple graph on n vertices with r colors which contain no monochromatic copy of Kk. It is shown that for every fixed k
Rainbow Turán Problems
The rainbow Turán problem for even cycles is studied, and the bound $\ex^*(n,C_6) = O(n^{4/3})$, which is of the correct order of magnitude, is proved.
Dirac's theorem for random graphs
Motivated by the study of resilience of random graph properties, it is proved that if p ≫ log n/n, then a.s. every subgraph of G(n,p) with minimum degree at least (1/2 + o (1) )np is Hamiltonian.
Acyclic edge colorings of graphs
A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an