3 problems

Let $G$ be an $n$-vertex Dirac graph, meaning that $\delta(G) \geq n/2$. Suppose every Dirac graph contains at least $c^n n!$ Hamilton cycles for some small positive constant $c$.…

Let $c$ satisfy $1/2<c\leqslant1$, and let $G$ be an $n$-vertex graph with minimum degree $\delta(G)\geqslant cn$. Colourful tree embedding conjecture. There exists a constant…

Let $c$ satisfy $1/2<c\leqslant1$, let $G$ be an $n$-vertex graph with minimum degree $\delta(G)\geqslant cn$, and suppose the edges of $G$ are properly edge-coloured. Colourful Ha…