Nemanja Škorić

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

A tight Erdős–Pósa function for long cycles

k \in \mathbb{N}

A General Lower Bound for Collaborative Tree Exploration

there exists

Almost-spanning universality in random graphs (Extended abstract)

C > 0

An algorithmic framework for obtaining lower bounds for random Ramsey problems

such that if

Powers of cycles in random graphs and hypergraphs

p^k \ge C \log^8 n / n

A tight Erd\H{o}s-P\'osa function for long cycles

then asymptotically almost surely the random graph