Related papers: On Pure Degree Sequence Manipulations Forcing Long…

We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.

For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in…

Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. Assuming rational independence of edge lengths, necessary and sufficient…

A Berge cycle of length $k$ in a hypergraph $\mathcal H$ is a sequence of distinct vertices and hyperedges $v_1,h_1,v_2,h_2,\dots,v_{k},h_k$ such that $v_{i},v_{i+1}\in h_i$ for all $i\in[k]$, indices taken modulo $k$. F\"uredi, Kostochka…

Let $D$ be a strongly connected directed graph of order $n\geq 4$ which satisfies the following condition (*): for every pair of non-adjacent vertices $x, y$ with a common in-neighbour $d(x)+d(y)\geq 2n-1$ and $min \{ d(x), d(y)\}\geq n-1$.…

We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{(\delta\geq1)}$. We prove that if $m=\tfrac n2(\log…

Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph…

We study stochastic processes that generate non-growing complex networks without self-loops and multiple edges (simple graphs). The work concentrates on understanding and formulation of constraints which keep the rewiring stochastic…

Dirac's classical theorem asserts that, for $n \ge 3$, any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian. Furthermore, if we additionally assume that such graphs are regular, then, by the breakthrough work of Csaba,…

Let $D$ be a digraph on $p\geq 5$ vertices with minimum degree at least $p-1$ and with minimum semi-degree at least $p/2-1$. For $D$ (unless some extremal cases) we present a detailed proof of the following results [12]: (i) $D$ contains…

A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We…

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations.…

We give an upper bound on the number of cycles in a simple graph in terms of its degree sequence, and apply this bound to resolve several conjectures of Kir\'aly and Arman and Tsaturian and to improve upper bounds on the maximum number of…

In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree…

In this paper we consider the existence of Hamilton cycles in the random graph $G=G_{n,m}^{\delta\geq 3}$. This a random graph chosen uniformly from the set of graphs with vertex set $[n]$, $m$ edges and minimum degree at least 3. Our…

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result…

We revisit the method of small subgraph conditioning, used to establish that random regular graphs are Hamiltonian a.a.s. We refine this method using new technical machinery for random $d$-regular graphs on $n$ vertices that hold not just…

We reconsider density matrices of graphs as defined in [quant-ph/0406165]. The density matrix of a graph is the combinatorial laplacian of the graph normalized to have unit trace. We describe a simple combinatorial condition (the "degree…

In this paper we focus on the problem of the degree sequence for the following random graph process. At any time-step $t$, one of the following three substeps is executed: with probability $\alpha_1$, a new vertex $x_t$ and $m$ edges…