Strongly light subgraphs in the 1-planar graphs with minimum degree 7

A graph is {\em $1$-planar} if it can be drawn in the plane such that every edge crosses at most one other edge. A connected graph $H$ is {\em strongly light} in a family of graphs $\mathfrak{G}$, if there exists a constant $\lambda$, such that every graph $G$ in $\mathfrak{G}$ contains a subgraph $K$ isomorphic to $H$ with $\deg_{G}(v) \leq \lambda$ for all $v \in V(K)$. In this paper, we present some strongly light subgraphs in the family of $1$-planar graphs with minimum degree~$7$.
Comment: 6 pages, 6 figures, http://amc-journal.eu/index.php/amc/article/view/564. in Ars Mathematica Contemporanea, 2015