  Given a collection $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs on the
common vertex set $V$ of size $n$, an $m$-edge graph $H$ on the same vertex set
$V$ is transversal in $\mathcal{G}$ if there exists a bijection $\varphi
:E(H)\rightarrow [m]$ such that $e \in E(G_{\varphi(e)})$ for all $e\in E(H)$.
Denote $\delta(\mathcal{G}):=\operatorname*{min}\left\{\delta(G_i): i\in
[m]\right\}$. In this paper, we first establish a minimum degree condition for
the existence of transversal Hamilton paths in $\mathcal{G}$: if $n=m+1$ and
$\delta(\mathcal{G})\geq \frac{n-1}{2}$, then $\mathcal{G}$ contains a
transversal Hamilton path. This solves a problem proposed by [Li, Li and Li, J.
Graph Theory, 2023]. As a continuation of the transversal version of Dirac's
theorem [Joos and Kim, Bull. Lond. Math. Soc., 2020] and the stability result
for transversal Hamilton cycles [Cheng and Staden, arXiv:2403.09913v1], our
second result characterizes all graph collections with minimum degree at least
$\frac{n}{2}-1$ and without transversal Hamilton cycles. We obtain an analogous
result for transversal Hamilton paths. The proof is a combination of the
stability result for transversal Hamilton paths or cycles, transversal blow-up
lemma, along with some structural analysis.
