For k ≥ 2 and r 1 such that + 4, we prove that, for any α > 0, there exists ε 0 the union of an n ‐vertex ‐graph with minimum codegree a binomial random on same vertex set contains th power tight Hamilton cycle high probability. This result = was first proved by McDowell Mycroft.

ABSTRACT We prove that in any ‐vertex complete graph, there is a collection of paths strongly separates pair distinct edges , meaning path which contains but not . Furthermore, for certain classes ‐regular graphs, we find edges. Both results are best‐possible up to the term.

We prove that for all $k\geq 4$ and $1\leq\ell<k/2$, every $k$-uniform hypergraph ${\mathcal{H}}$ on $n$ vertices with $\delta_{k-2}({\mathcal{H}})\geq(\tfrac{4(k-\ell)-1}{4(k-\ell)^2}+o(1))\binom{n}{2}$ contains a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$. This degree condition is asymptotically best possible. The case $k=3$ was addressed earlier by Buß et al.

Abstract Given graphs and a positive integer , say that is ‐ Ramsey for denoted if every ‐coloring of the edges contains monochromatic copy . The size‐Ramsey number graph defined to be Answering question Conlon, we prove that, fixed have where th power ‐vertex path (ie, with vertex set all such distance between in at most ). Our proof probabilistic, but can also made constructive.

For graphs G and H, let G⟶prbH denote the property that, for every proper edge-colouring of (with an arbitrary number colours) there is a totally multicoloured, or rainbow, copy H in G, that is, with no two edges same colour. We consider problem establishing threshold pHrb=pHrb(n) this binomial random graph G(n,p). More specifically, we give upper bound pHrb extend our result to certain locally bounded colourings generalize colourings. Our method heavily based on characterization sparse...

Abstract For graphs G and H , let denote the property that for every proper edge‐coloring of (with an arbitrary number colors) there is a rainbow copy in is, with no two edges same color. The authors (2014) proved that, graph threshold function this binomial random asymptotically at most where denotes so‐called maximum 2‐density . Nenadov et al. if cycle least seven vertices or complete 19 vertices, then We show exists fairly rich, infinite family F containing triangle such suitable...

Given a positive integer s, the s-colour size-Ramsey number of graph H is smallest m such that there exists G with edges property that, in any colouring E ( ) s colours, monochromatic copy H. We prove for integers k and kth power n-vertex bounded degree tree linear n. As corollary, we obtain graphs treewidth n, which answers question raised by Kamčev, Liebenau, Wood Yepremyan.

Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning the threshold for following Ramsey-type property $G$: every $k$-colouring of edge set $G$ yields $k$ pairwise vertex disjoint trees that partition whole $G$. We determine this two colours.

For graphs $G,H$, we write $G \overset{\mathrm{rb}}{\longrightarrow} H $ if for every proper edge-coloring of $G$ there is a rainbow copy $H$, i.e., where no color appears more than once. Kohayakawa, Konstadinidis and the last author proved that threshold $G(n,p) H$ at most $n^{-1/m_2(H)}$. Previous results have matched lower bound this anti-Ramsey cycles complete with least 5 vertices. also presented an infinite family $H$ which asymptotically smaller In paper, devise framework provides...

Let $\vec H$ be an orientation of a graph $H$. Alon and Yuster proposed the problem determining or estimating $D(n,m,\vec H)$, maximum number H$-free orientations with $n$ vertices $m$ edges may have. We consider typical graphs $G(n,m)$ edges. Suppose H =C^\circlearrowright_\ell $ is directed cycle length $\ell\geq 3$. show that if ${m\gg n^{1+1/(\ell-1)}}$, then this $2^{o(m)}$, while ${m\ll it $2^{(1-o(1))m}$.

Dados grafos G e H, denotamos a seguinte propriedade por rb → p H: para toda coloração própria das arestas de (com uma quantidade arbitrária cores) existe cópia multicolorida H em G, i.e., sem duas da mesma cor. Sabe-se que, todo grafo função limiar pHrb = pHrb(n) essa no aleatório binomial G(n, p) é assintoticamente máximo n-1/m(2)(H), onde m(2)(H) denota assim chamada 2-densidade máxima H. Neste trabalho discutimos esse alguns resultados recentes estudo propriedades anti-Ramsey aleatórios,...

We prove for all k\geq 4 and 1\leq \ell &lt;k/2 the sharp minimum (k-2)-degree bound a k-uniform hypergraph H on n vertices to contain Hamiltonian \ell-cycle if k-\ell divides is sufficiently large. This extends result of Han Zhao 3-uniform hypegraphs.