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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 study conjectures relating degree conditions in $3$-partite hypergraphs to the matching number of the hypergraph, and use topological methods to prove special cases. In particular, we prove a strong version of a theorem of Drisko…

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…

Balogh, Csaba, Jing and Pluh\'ar recently determined the minimum degree threshold that ensures a $2$-coloured graph $G$ contains a Hamilton cycle of significant colour bias (i.e., a Hamilton cycle that contains significantly more than half…

Let $G$ be a graph on an even number $n$ of vertices and let ${\cal M}_G$ be the collection of perfect matchings in $G$. Dirac's theorem says that if the minimum degree $\delta(G)$ of $G$ is at least $n/2$, then ${\cal M}_G$ is guaranteed…

The famous Dirac's Theorem gives an exact bound on the minimum degree of an $n$-vertex graph guaranteeing the existence of a hamiltonian cycle. We prove exact bounds of similar type for hamiltonian Berge cycles in $r$-uniform, $n$-vertex…

A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $\lceil n/2 \rceil$ is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of…

A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges…

The conflict-free chromatic index of a graph $G$ is the minimum number of colours in an edge colouring of $G$ such that the neighbourhood of every edge contains a colour appearing exactly once. Its vertex analogue is the conflict-free…

Dirac's theorem (1952) is a classical result of graph theory, stating that an $n$-vertex graph ($n \geq 3$) is Hamiltonian if every vertex has degree at least $n/2$. Both the value $n/2$ and the requirement for every vertex to have high…

A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured that every graph with minimum degree at least 2n/3 contains the square of a Hamiltonian…

We investigate Hamilton cycles in edge-colored graphs with \( r \) colors, focusing on the notion of color-bias (discrepancy), the maximum deviation from uniform color frequencies along a cycle. Foundational work by Balogh, Csaba, Jing, and…

Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering…

A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an…

Given a graph $G$ and an $r$-edge-colouring $\chi$ on $E(G)$, a Hamilton cycle $H\subset G$ is said to have $t$ colour-bias if $H$ contains $n/r+t$ edges of the same colour in $\chi$. Freschi, Hyde, Lada and Treglown showed every…

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…

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this…

We show the existence of rainbow perfect matchings in $\mu n$-bounded edge colourings of Dirac bipartite graphs, for a sufficiently small $\mu>0$. As an application of our results, we obtain several results on the existence of rainbow…

In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for $k\ge 3$, if $H$ is a $k$-graph on $n\in k\mathbb N$ vertices with independence number at most $n/p$ and minimum codegree at least…

We show that there is a constant $C$ such that for every $\varepsilon>0$ any $2$-coloured $K_n$ with minimum degree at least $n/4+\varepsilon n$ in both colours contains a complete subgraph on $2t$ vertices where one colour class forms a…