Related papers: Compatible Powers of Hamilton Cycles in Dense Grap…

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ whenever $G$ is an $F$-free graph of maximum degree $\Delta$. The…

Let $k,a,b$ be positive integers with $a+b=k$. A $k$-uniform hypergraph is called an $(a,b)$-cycle if there is a partition $(A_0,B_0,A_1,B_1,\ldots,A_{t-1},B_{t-1})$ of the vertex set with $|A_i|=a$, $|B_i|=b$ such that $A_i\cup B_i$ and…

The aim of this paper is to extend and generalise some work of Katona on the existence of perfect matchings or Hamilton cycles in graphs subject to certain constraints. The most general form of these constraints is that we are given a…

In 1973, Chv\'atal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. This conjecture has inspired extensive research and has been verified for several special classes…

Let $\mathbf{G}=\{G_1, \ldots, G_m\}$ be a graph collection on a common vertex set $V$ of size $n$ such that $\delta(G_i) \geq (1+o(1))n/2$ for every $i \in [m]$. We show that $\mathbf{G}$ contains every Hamilton cycle pattern. That is, for…

In this paper, we investigate the Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform (DFT) in a $d$ dimensional system. The uncertainty diagram of complete incompatibility bases $\mathcal {A},\mathcal…

Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $\delta(H_n)\geq\alpha n$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will…

We use a randomised embedding method to prove that for all \alpha>0 any sufficiently large oriented graph G with minimum in-degree and out-degree \delta^+(G),\delta^-(G)\geq (3/8+\alpha)|G| contains every possible orientation of a Hamilton…

The bipartite independence number of a graph $G$, denoted as $\tilde\alpha(G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two sets $A,B\subseteq V(G)$ with…

In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a…

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to…

The Hamiltonian cycle (HC) problem in graph theory is a well-known NP-complete problem. We present an approach in terms of $\mathbb{Z}_2$ lattice gauge theory (LGT) defined on the lattice with the graph as its dual. When the coupling…

A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$…

We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$-uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to R\"odl, Ruci\'nski, and Szemer\'edi. For graphs, Chv\'atal…

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains…

The anti-forcing number of a perfect matching $M$ of a graph $G$ is the minimum number of edges of $G$ whose deletion results in a subgraph with a unique perfect matching $M$, denoted by $af(G,M)$. When $G$ is a plane bipartite graph, Lei…

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich,…

A Hamiltonian path (cycle) in a graph is a path (cycle, respectively) which passes through all of its vertices. The problems of deciding the existence of a Hamiltonian cycle (path) in an input graph are well known to be NP-complete, and…

The P\'osa-Seymour conjecture asserts that every graph on $n$ vertices with minimum degree at least $(1 - 1/(r+1))n$ contains the $r^{th}$ power of a Hamilton cycle. Koml\'os, S\'ark\"ozy and Szemer\'edi famously proved the conjecture for…

For $1\le d\le \ell< k$, we give a new lower bound for the minimum $d$-degree threshold that guarantees a Hamilton $\ell$-cycle in $k$-uniform hypergraphs. When $k\ge 4$ and $d< \ell=k-1$, this bound is larger than the conjectured minimum…