An extension of the blow-up lemma to arrangeable graphs

The blow-up lemma established by Komlós, Sárközy, and Szemerédi in 1997 is an important tool for the embedding of spanning subgraphs of bounded maximum degree. Here we prove several generalizations of this result concerning the embedding of $a$-arrangeable graphs, where a graph is called $a$-arrangeable if its vertices can be ordered in such a way that the neighbors to the right of any vertex $v$ have at most $a$ neighbors to the left of $v$ in total. Examples of arrangeable graphs include planar graphs and, more generally, graphs without a $K_s$-subdivision for constant $s$. Our main result shows that $a$-arrangeable graphs with maximum degree at most $\sqrt{n}/\log n$ can be embedded into corresponding systems of superregular pairs. This is optimal up to the logarithmic factor. We also present two applications. We prove that any large enough graph $G$ with minimum degree at least $\big(\frac{r-1}{r}+\gamma\big)n$ contains an $F$-factor of every $a$-arrangeable $r$-chromatic graph $F$ with at most $\xi n$ vertices and maximum degree at most $\sqrt{n}/\log n$, as long as $\xi$ is sufficiently small compared to $\gamma/(ar)$. This extends a result of Alon and Yuster [J. Combin. Theory Ser. B, 66 (1996), pp. 269--282]. Moreover, we show that for constant $p$ the random graph $\mathcal{G}(n,p)$ is universal for the class of $a$-arrangeable $n$-vertex graphs $H$ of maximum degree at most $\xi n/\log n$, as long as $\xi$ is sufficiently small compared to $p/a$.