Bandwidth, treewidth, separators, expansion, and universality

We prove that planar graphs with bounded maximum degree have sublinear bandwidth. As a consequence for each γ > 0 every n-vertex graph with minimum degree ( 4 + γ)n contains a copy of every bounded-degree planar graph on n vertices. The proof relies on the fact that planar graphs have small separators. Indeed, we show more generally that for any class of bounded-degree graphs the concepts of sublinear bandwidth, sublinear treewidth, the absence of big expanders as subgraphs, and the existence of small separators are equivalent. 1 Email: {boettche|taraz}@ma.tum.de, [email protected] 2 Email: [email protected] 1 Bandwidth and universality for planar graphs Let G = (V,E) be a graph on n = |V | vertices. The bandwidth of G is denoted by bw(G) and defined to be the minimum b ∈ N, such that there exists a labelling of the vertices in V by numbers 1, . . . , n so that the labels of adjacent vertices differ by at most b. In [2] Chung proved that any tree T with n vertices and maximum degree ∆ has bandwidth at most 5n/ log∆(n), and it is easy to see that this bound is sharp up to the multiplicative constant. Our first theorem extends Chung’s result to planar graphs. Theorem 1.1 Let G be a planar graph on n vertices with maximum degree ∆. Then the bandwidth of G is bounded from above by bw(G) ≤ 15n log∆(n) . Our main motivation for studying the bandwidth is that it turns out to be helpful when embedding a bounded-degree graphs G into a graph H with sufficiently high minimum degree, even when G and H have the same number of vertices. Dirac’s theorem [4] concerning the existence of Hamiltonian cycles in graphs of minimum degree n/2 is a classical example for theorems of this type. It was followed by results of Corrádi and Hajnal [3], Hajnal and Szemerédi [5] about embedding Kr-factors, and more recently by a series of theorems due to Komlós, Sarközy, and Szemerédi and others (see e.g. [7,8,9]) which deal with powers of Hamiltonian cycles, trees, and H-factors. Along the lines of these results the following unifying theorem was conjectured by Bollobás and Komlós [6] and recently proven by Böttcher, Schacht, and Taraz [1]. Theorem 1.2 For all r,∆ ∈ N and γ > 0, there exist constants β > 0 and n0 ∈ N such that for every n ≥ n0 the following holds. If G is an r-chromatic graph on n vertices with ∆(G) ≤ ∆ and bandwidth at most βn and if H is a graph on n vertices with minimum degree δ(H) ≥ ( r−1 r + γ)n, then G can be embedded into H. Combining Theorems 1.1 and 1.2 immediately yields the following result which states that all sufficiently large graphs with minimum degree ( 4 + γ)n are universal for the class of bounded-degree planar graphs. Corollary 1.3 For all ∆ ∈ N and γ > 0, there exists n0 ∈ N such that for every n ≥ n0 the following holds: The first and third author were partially supported by DFG grant TA 309/2-1. The first author was partially supported by a Minerva grant. (a ) Every 3-chromatic planar graph on n vertices with maximum degree at most ∆ can be embedded into every graph on n vertices with minimum degree at least ( 3 + γ)n. (b ) Every planar graph on n vertices with maximum degree at most ∆ can be embedded into every graph on n vertices with minimum degree at least ( 4 + γ)n. This extends a result by Kühn, Osthus, and Taraz [10], who proved that for every graph H with minimum degree at least ( 3 +γ)n there exists a particular spanning triangulation G that can be embedded into H . 2 Expansion, separators, and treewidth In the first section we observed the connection between results about the bandwidth of a class of graphs and embedding problems due to Theorem 1.2. This raises the question which rôle the bandwidth plays in this theorem. It is not difficult to see that the condition on the bandwidth of G in Theorem 1.2 is necessary already for the case r = 2: Let G be a random bipartite graph with bounded maximum degree and letH be the graph formed by two cliques of size (1/2 + γ)n each, which share exactly 2γn vertices. Then H cannot contain a copy of G, since in G every vertex set of size (1/2 − γ)n has more than 2γn neighbours. The reason for this obstacle is that H has good expansion properties. In light of Theorem 1.2 the same example shows that graphs with sublinear bandwidth (as in Theorem 1.1) cannot exhibit good expansion properties (definitions and exact statements are provided below). One may ask whether the converse is also true, i.e. whether bad expansion properties in bounded-degree graphs lead to small bandwidth. We will show that this is indeed the case via the existence of certain separators. A similar approach can be used to prove Theorem 1.1. In fact, we will show a more general theorem (Theorem 2.4) which proves that the concepts of sublinear bandwidth, sublinear treewidth, bad expansion properties, and sublinear separators are equivalent for graphs of bounded maximum degree. Since planar graphs have sublinear separators [11] Theorem 1.1 is a direct consequence of this theorem. For the precise statement of Theorem 2.4, we need the following definitions. We start with the notions of tree decompositions and treewidth. Roughly speaking, a tree decomposition tries to arrange the vertices of a graph in a tree-like manner and the treewidth measures how well this can be done. Definition 2.1 (treewidth) A tree decomposition of a graph G = (V,E) is a pair ({Xi : i ∈ I}, T = (I, F )) where {Xi : i ∈ I} is a family of subsets Xi ⊆ V with ⋃ i∈I Xi = V , and where T = (I, F ) is a tree such that for every edge {v, w} ∈ E there exists i ∈ I with {v, w} ⊆ Xi and for every i, j, k ∈ I such that j lies on the path from i to k in T we have Xi ∩Xk ⊆ Xj. The width of ({Xi : i ∈ I}, T = (I, F )) is defined as maxi∈I |Xi| − 1. The treewidth tw(G) of G is the minimum width of a tree decomposition of G. A vertex set is said to be expanding, if it has many external neighbours. We call a graph bounded, if every sufficiently large subgraph contains a subset which is not expanding. Definition 2.2 (bounded) Let 0 < ε < 1 be a real number, b ∈ N and G = (V,E) be a graph. G is called (b, ε)-bounded, if for every subgraph G ⊆ G with |V (G′)| ≥ b vertices there exists a subset U ⊆ V (G) such that |U | ≤ |V (G′)|/2 and |N(U)| ≤ ε|U |. (Here N(U) is the set of neighbours of vertices in U that lie outside of U .) Finally, a separator in a graph is a small cut-set that splits the graph into components of limited size. Definition 2.3 (separator) Let 0 < α < 1 be a real number, b ∈ N and G = (V,E) a graph. A subset S ⊆ V is said to be a (b, α)-separator of G, if there exist subsets A,B ⊆ V such that V = A∪̇B∪̇S, |S| ≤ b, |A|, |B| ≤ α|V (G)|, and E(A,B) = ∅, where E(A,B) denotes the set of edges with one end in A and one in B. With these definitions at hand, we are ready to state our main theorem. Theorem 2.4 Let ∆ be an arbitrary but fixed positive integer and consider a class of graphs C such that all graphs in C have maximum degree at most ∆. Denote by Cn the set of those graphs in C with n vertices. Then the following four properties are equivalent: (1 ) For every β1 > 0 there exists n0 such that for all n ≥ n0 every graph in Cn has treewidth at most β1n. (2 ) For every β2 > 0 there exists n0 such that for all n ≥ n0 every graph in Cn has bandwidth at most β2n. (3 ) For every β3 > 0 and every ε > 0 there exists n0 such that for all n ≥ n0 every graph in Cn is (β3n, ε)-bounded. (4 ) For every β4 > 0 there exists n0 such that for all n ≥ n0 every graph in Cn has a (β4n, 2/3)-separator. If the class C meets one (and thus all) of the above conditions, then the following is also true. (5 ) For every γ > 0 and r ∈ N there exists n0 such that for all n ≥ n0 and for every graph G ∈ Cn with chromatic number r and for every graph H on n vertices with minimum degree at least ( r−1 r + γ)n, the graph H contains a copy of G.