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Finding a large hidden clique in a random graph
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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