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 $p/a$.

We prove that for any pair of constants a>0 and D and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most D, and with at most n(n-1)/2 edges in total packs into the complete graph of order (1+a)n. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

This paper extends the scenario of the Four Color Theorem in the following way. Let [Formula: see text] be the set of all [Formula: see text]-uniform hypergraphs that can be (linearly) embedded into [Formula: see text]. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in [Formula: see text]. For example, we can prove that for [Formula: see text] there are hypergraphs in [Formula: see text] on [Formula: see text] vertices whose chromatic number is [Formula: see text], whereas the chromatic number for [Formula: see text]-vertex hypergraphs in [Formula: see text] is bounded by [Formula: see text] for [Formula: see text].

The conjecture of Bollobás and Komlós, recently proved by Böttcher, Schacht, and Taraz [Math. Ann., 343 (2009), pp. 175–205], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph G with minimum degree $(1+\gamma)n$, provided that n is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac{1}{2}+\gamma)n$ when we have the additional structural information of the host graph G being balanced bipartite. This complements results of Zhao [SIAM J. Discrete Math., 23 (2009), pp. 888–900], as well as Hladký and Schacht [SIAM J. Discrete Math., 24 (2010), pp. 357–362], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with r and s fixed. Moreover, our result can be used to prove that in every balanced bipartite graph G on $2n$ vertices with minimum degree $(\frac{1}{2}+\gamma)n$ and n sufficiently large, the set of Hamilton cycles of G is a generating system for its cycle space.

We study the evolution of the chromatic number of a random intersection graph and show that, in a certain range of parameters, these random graphs can be colored optimally with high probability using different greedy algorithms. Experiments on real network data confirm the positive theoretical predictions and suggest that heuristics for the clique and the chromatic number can work hand in hand proving mutual optimality.