Rajko Nenadov

A Short Proof of the Random Ramsey Theorem

In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r ≥ 2, there exist constants c, C > 0 such that P[Gn,p → (F )r] = { 1− o(1), p ≥ Cn−1/m2(F ) o(1), p ≤ cn−1/m2(F , where m2(F ) = maxJ⊆F,vJ≥2 eJ−1 vJ−2 . The proof of the 1-statement is based on the recent beautiful hypergraph container theorems by Saxton/Thomason and Balogh/Morris/Samotij. The proof of the 0-statement is elementary.

Universality of random graphs and rainbow embedding

In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of . Let the maximum density of a graph H be the maximum average degree of all the subgraphs of H. First, we show that for , a graph w.h.p. contains copies of all spanning graphs H with maximum degree at most Δ and maximum density at most d. For , this improves a result of Dellamonica, Kohayakawa, Rodl and Rucincki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p. all such graphs for . In particular, if , the random graph therefore contains w.h.p. every spanning tree with maximum degree bounded by Δ. This improves a result of Johannsen, Krivelevich and Samotij. Finally, in the same spirit, we show that for any spanning graph H with constant maximum degree, and for suitable p, if we randomly color the edges of a graph with colors, then w.h.p. there exists a rainbow copy of H in G (that is, a copy of H with all edges colored with distinct colors).

Robust hamiltonicity of random directed graphs

Abstract In his seminal paper from 1952 Dirac showed that the complete graph on n ≥ 3 vertices remains Hamiltonian even if we allow an adversary to remove ⌊ n / 2 ⌋ edges touching each vertex. In 1960 Ghouila–Houri obtained an analogue statement for digraphs by showing that every directed graph on n ≥ 3 vertices with minimum in- and out-degree at least n / 2 contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of local resilience. The local resilience of a graph (digraph) G with respect to a property P is the maximum number r such that G has the property P even if we allow an adversary to remove an r-fraction of (in- and out-going) edges touching each vertex. The theorems of Dirac and Ghouila–Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is 1/2. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability p = ω ( log ⁡ n / n ) with respect to Hamiltonicity is 1 / 2 ± o ( 1 ) . For random directed graphs, Hefetz, Steger and Sudakov (2014) proved an analogue statement, but only for edge probability p = ω ( log ⁡ n / n ) . In this paper we significantly improve their result to p = ω ( log 8 ⁡ n / n ) , which is optimal up to the polylogarithmic factor.

Almost-spanning universality in random graphs

A graph G is said to be i¾?n,Δ-universal if it contains every graph on at most n vertices with maximum degree at most Δ. It is known that for any e>0 and any natural number Δ there exists c>0 such that the random graph Gn, p is asymptotically almost surely i¾?1-en,Δ-universal for pi¾?clogn/n1/Δ. Bypassing this natural boundary, we show that for Δi¾?3 the same conclusion holds when pi¾?n-1Δ-1log5n.

On finite reflexive homomorphism-homogeneous binary relational systems

A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P.?J.?Cameron and J.?Nesetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely induced substructures of the structure extends to an endomorphism of the structure.In this paper, we consider finite homomorphism-homogeneous relational systems with one reflexive binary relation. We show that for a large part of such relational systems (bidirectionally connected digraphs; a digraph is bidirectionally connected if each of its connected components can be traversed by ? -paths) the problem of deciding whether the system is homomorphism-homogeneous is coNP-complete. Consequently, for this class of relational systems there is no polynomially computable characterization (unless P = N P ). On the other hand, in case of bidirectionally disconnected digraphs we present the full characterization. Our main result states that if a digraph is bidirectionally disconnected, then it is homomorphism-homogeneous if and only if it is either a finite homomorphism-homogeneous quasiorder, or an inflation of a homomorphism-homogeneous digraph with involution (a specific class of digraphs introduced later in the paper), or an inflation of a digraph whose only connected components are C 3 ? and? 1 ? .

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

We show that for every

On the threshold for the Maker‐Breaker H‐game

k \in \mathbb{N}

On the number of graphs without large cliques

there exists

Star-factors in graphs with large minimum degree

C > 0

Optimal induced universal graphs for bounded-degree graphs

such that if