We give a simple short proof of Brooks' theorem using only induction and greedy coloring, while avoiding issues of graph connectivity. The argument generalizes easily to some extensions of Brooks' theorem, including its variants for list coloring, signed graphs coloring and correspondence coloring.

We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nešetřil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion.More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings.

An immersion of a graph H into a graph G is a one-to-one mapping f : V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f (u) and f (v). The immersion is strong if the paths P uv are internally disjoint from f (V (H)). It is proved that for every positive integer t, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2 , where c > 0 is an absolute constant. For small values of t, 1 ≤ t ≤ 7, every simple graph of minimum degree at least t − 1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

The star chromatic index χs′(G) of a graph G is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi‐colored. We obtain a near‐linear upper bound in terms of the maximum degree Δ=Δ(G). Our best lower bound on χnormals′ in terms of Δ is 2Δ(1+o(1)) valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.

A result of Plotkin, Rao, and Smith implies that graphs with polynomial expansion have strongly sublinear separators. We prove a converse of this result showing that hereditary classes of graphs with strongly sublinear separators have polynomial expansion. This confirms a conjecture of the first author. *

Let G be a graph with a list assignment L. Suppose a preferred color is given for some of the vertices; how many of these preferences can be respected when L-coloring G? We explore several natural questions arising in this context, and propose directions for further research.

We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series. *

Let hom (G, H) be the number of homomorphisms from a graph G to a graph H. A well-known result of Lovász states that the function hom (·, H) from all graphs uniquely determines the graph H up to isomorphism. We study this function restricted to smaller classes of graphs. We show that several natural classes (2-degenerate graphs and graphs homomorphic to an arbitrary non-bipartite graph) are sufficient to recognize all graphs, and provide a description of graph properties that are recognizable by graphs with bounded tree-width. ᭧