Jekyll2023-01-08T18:10:24+00:00https://minorfree.github.io/feed.xmlRambling on GraphsBlog by Hung LeHung LeShortcutting Trees2023-01-08T00:00:00+00:002023-01-08T00:00:00+00:00https://minorfree.github.io/tree-shortcutting<p>Over the past two years or so, I have been thinking about a cute problem, which turned out to be much more useful to my research than I initially thought. Here it is:</p> <hr /> <p><strong>Tree Shortcutting Problem</strong>: Given an edge-weighted tree $$T$$, add (weighted) edges to $$T$$, called <em>shortcuts</em>, to get a graph $$K$$ such that:</p> <ol> <li>$$d_K(u,v) = d_T(u,v)~\quad \forall u,v\in V(T)$$. That is, $$K$$ preserves distances in $$T$$.</li> <li>For every $$u,v\in V(T)$$, there exists a shortest path from $$u$$ to $$v$$ in $$K$$ containing at most $$k$$ edges for some $$k\geq 2$$.</li> </ol> <p>The goal is to minimize the product $$k \cdot \mathrm{tw}(K)$$, where $$\mathrm{tw}(K)$$ is the treewidth of $$K$$.</p> <hr /> <p><img src="/assets/figs/shortcut.svg" alt="" /></p> <p><em>Figure 1: An emulator $$K$$ obtained by adding one edge to the tree $$T$$ has hop bound $$3$$ and treewidth $$2$$. Compared to $$T$$, the hop bound decreases by 1 while the treewidth increases by 1.</em></p> <p>Such a graph $$K$$ is called a <em>low-hop and low-treewidth emulator</em> of the tree $$T$$. The parameter $$k$$ is called the <em>hop bound</em> of $$K$$. It is expected that there will be some trade-off between the hop bound and the treewidth. We are interested in minimizing $$k \cdot \mathrm{tw}(K)$$. This product directly affects parameters in our application; see the conclusion section for more details.</p> <p>For readers who are not familar with treeewidth, see <a href="https://en.wikipedia.org/wiki/Treewidth">here</a> and <a href="https://www.win.tue.nl/~nikhil/courses/2015/2WO08/treewidth-erickson.pdf">here</a> for an excellent introduction, and why treewidth is an interesting graph parameter.</p> <blockquote> <p><strong>Remark 1:</strong> The tree shortcutting problem is already non-trivial for unweighted trees; the shortcuts must be weighted though. Furthermore, for each edge $$(u,v)$$ added to $$K$$, the weight of the edge will be $$d_T(u,v)$$. Thus, in the construction below, we do not explicitly assign weights to the added edges.</p> </blockquote> <p>A version of a tree shortcutting problem where one seeks to minimize <em>the number of edges of $$K$$</em>, given a hop bound $$k$$, was studied extensively (see <a href="http://www.math.tau.ac.il/~haimk/adv-ds-2008/Alon-Schieber.ps">1</a>,<a href="https://dl.acm.org/doi/abs/10.5555/640186.640193">2</a>,<a href="https://arxiv.org/abs/1005.4155">3</a>,<a href="https://dl.acm.org/doi/10.1145/800070.802185">4</a>, including my own work <a href="https://arxiv.org/abs/2107.14221">with</a> <a href="https://arxiv.org/abs/2112.09124">others</a>), arising in the context of spanners and minimum spanning tree problem. What is interesting there IMO is that we see all kinds of crazy slowly growing functions in computer science: for $$k = 2$$, the number of edges of $$K$$ is $$\Theta(n \log n)$$; for $$k = 3$$, the number of edges of $$K$$ is $$\Theta(n \log\log n)$$; for $$k = 4$$, the number of edges is $$\Theta(n \log^* n)$$; $$\ldots$$ [too difficult to describe]; and for $$k = \alpha(n)$$, the number of edges is $$\Theta(n)$$. Here, as you might guess, $$\alpha(\cdot)$$ is the notorious (one parameter) inverse Ackermann function. (The $$\Theta$$ notation in the number of edges means there exist matching lower bounds.) I hope to cover this problem in a future blog post.</p> <p>Now back to our tree shortcutting problem. Let $$n = \lvert V(T) \rvert$$. There are two extreme regimes that I am aware of:</p> <ol> <li>Hop bound $$k=2$$ and treewidth $$\mathrm{tw}(K) = O(\log n)$$. This is a relatively simple exercise.</li> <li>Hop bound $$k=O(\log n)$$ and treewidth $$\mathrm{tw}(K) = O(1)$$. This regime is harder to prove; trying to show this for a path graph will be an insightful exercise. It follows from a well-known fact  that any tree decomposition of width $$t$$ can be turned into a tree decomposition of width $$O(t)$$ and depth $$O(\log n)$$.</li> </ol> <p>The two regimes might suggest that a lower bound $$k\cdot \mathrm{tw}(K) = \Omega(\log n)$$ for any $$k$$. In our recent paper , we show that this is not the case:</p> <hr /> <p><strong>Theorem 1</strong>: There exists an emulator $$K$$ for any $$n$$-vertex tree $$T$$ such that $$h(K) = O(\log \log n)$$ and $$\mathrm{tw}(K) = O(\log \log n)$$.</p> <hr /> <p>Theorem 1 implies that one can get an emulator with $$k\cdot \mathrm{tw}(K) = O((\log \log n)^2)$$, which is exponentially smaller than $$O(\log(n))$$. The goal of this post is to discuss the proof of Theorem 1. See the conclusion section for a more thorough discussion on other aspects of Theorem 1, in particular, the construction time and application.</p> <p>For the tree shortcutting problem, it is often insightful to look into the <strong>path graph with $$n$$ vertices</strong>, which is a special case. Once we solve the path graph, extending the ideas to trees is not that difficult.</p> <h1 id="1-the-path-graph">1. The Path Graph</h1> <p>The (unweighted )path graph $$P_n$$ is a path of $$n$$ vertices. To simplify the presentation, assume that $$\sqrt{n}$$ is an integer. The construction is recursive and described in the pseudo-code below. First we divide $$P_n$$ into $$\sqrt{n}$$ sub paths of size $$\sqrt{n}$$ each. Denote the endpoints of these subpaths by $$b_i = i\sqrt{n}$$ for $$1\leq i\leq\sqrt{n}$$. We call $${b_i}_{i}$$ boundary vertices. We have two types of recursions: (1) top level recursion – lines 2 and 3 – and (2) subpath recursion – lines 5 to 9. The intuition of the top level recursion is a bit tricky and we will get to that later. See Figure 2.</p> <p><img src="/assets/figs/recursion.svg" alt="" /></p> <p><em>Figure 2: (a) The recursive construction applied to the path $$P_n$$. (b) Gluing $$\mathcal T_B$$ and all $$\{\mathcal T_i\}$$ to obtain a tree decomposition $$\mathcal{T}$$ of $$K$$ via red edges.</em></p> <p>The subpath recursion is natural: recursively shortcut each subpath $$P[b_i,b_{i+1}]$$ (line 6). The subpath recursion returns the shortcut graph $$K_i$$ and its tree decomposition $$\mathcal T_i$$. Next, we add edges from each boundary vertex $$b_i, b_{i+1}$$ of the subpath to all other vertices on the subpath (lines 7 and 8). This step guarantees that each vertex of the subpath can “jump” to the boundary vertices using only <strong>one edge</strong>. In terms of tree decomposition, it means that we add both $$b_i$$ and $$b_{i+1}$$ to every bag of $$\mathcal T_i$$ (line 9).</p> <p>The top level recursion serves two purposes: (i) creating a low hop emulator for boundary vertices (recall that each vertex can jump to a boundary vertex in the same subpath using one edge) and (ii) gluing $${\mathcal T_i}$$ together. More precisely, let $$P_{\sqrt{n}}$$ be a path of boundary vertices, i.e., $$b_i$$ is adjacent to $$b_{i+1}$$ in $$P_{\sqrt{n}}$$. We shortcut $$P_{\sqrt{n}}$$ recursively, getting the shortcut graph $$K_B$$ and its tree decomposition $$\mathcal T_B$$ (line 3). Since $$(b_i,b_{i+1})$$ is an edge in $$P_{\sqrt{n}}$$, there must be a bag in $$\mathcal T_B$$ containing both $$b_i,b_{i+1}$$; that is, the bag $$X$$ in line 11 exists. See Figure 2(b). Recall that in line 9, every bag in $$\mathcal T_i$$ contains both $$b_i,b_{i+1}$$, and that $$K_B$$ and $$K_i$$ only share two boundary vertices $$b_i,b_{i+1}$$. Thus, we can connect $$X$$ to an arbitrary bag of $$\mathcal T_i$$ as done in line 12. This completes the shortcutting algorithm.</p> <hr /> <p><span style="font-variant: small-caps">PathShortcutting</span>$$(P_n)$$</p> <blockquote> <p>$$1.$$ $$B \leftarrow {0,\sqrt{n}, 2\sqrt{n}, \ldots, n}$$ and $$b_i \leftarrow i\sqrt{n}$$ for every $$0\leq i \leq \sqrt{n}$$<br /> $$2.$$ $$P_{\sqrt{n}} \leftarrow$$ unweighted path graph with vertex set $$B$$.<br /> $$3.$$ $$(K_B,\mathcal T_B) \leftarrow$$<span style="font-variant: small-caps">PathShortcutting</span>$$(P_{\sqrt{n}})$$<br /> $$4.$$ $$K\leftarrow K_B,\quad \mathcal{T}\leftarrow \mathcal T_B$$<br /> $$5.$$ for $$i\leftarrow 0$$ to $$\sqrt{n}-1$$<br /> $$6.$$     $$(K_i,\mathcal T_i) \leftarrow$$<span style="font-variant: small-caps">PathShortcutting</span>$$(P_{n}[b_i, b_{i+1}])$$<br /> $$7.$$      for each $$v\in P_{n}[b_i, b_{i+1}]$$<br /> $$8.$$           $$E(K_i)\leftarrow {(v,b_i), (v,b_{i+1})}$$ <br /> $$9.$$           add both $${v_i,v_{i+1}}$$ to every bag of $$\mathcal T_i$$<br /> $$10.$$     $$K\leftarrow K \cup K_i$$ <br /> $$11.$$     Let $$X$$ be a bag in $$\mathcal{T}$$ containing both $$b_i,b_{i+1}$$<br /> $$12.$$     Add $$\mathcal T_i$$ to $$\mathcal{T}$$ by connecting $$X$$ to an arbitrary bag of $$\mathcal T_i$$ <br /> $$13.$$ return $$(K,\mathcal{T})$$</p> </blockquote> <hr /> <p>It is not difficult to show that $$\mathcal{T}$$ indeed is a tree decomposition of $$K$$. Thus, we focus on analyzing the hop bound and the treewidth.</p> <blockquote> <p><strong>Remark 2:</strong> For notational convenience, we include $$0$$ in the set $$B$$ though $$0\not\in P_n$$. When calling the recursion, one could simply drop 0.</p> </blockquote> <p><img src="/assets/figs/hopbound.svg" alt="" /></p> <p><em>Figure 3: A low-hop path from $$u$$ to $$v$$.</em></p> <p><strong>Analyzing the hop bound $$h(K)$$.</strong> Let $$u$$ and $$v$$ be any two vertices of $$P_{n}$$; w.l.o.g, assume that $$u \leq v$$, and $$h(n)$$ be the hop bound. Let $$b_{u}$$ and $$b_v$$ be two boundary vertices of the subpaths containing $$u$$ and $$v$$, respectively, such that $$b_u,b_v \in P[u,v]$$. See Figure 3. As mentioned above, line 8 of the algorithms guarantees that there are two edges $$(u,b_u)$$ and $$(b_v,v)$$ in $$K$$, and the top level recursion (line 3) guarantees that there is a shortest path of hop length $$h(\sqrt{n})$$ between $$b_u$$ and $$b_v$$ in $$K_B$$. Thus we have:</p> $h(n) \leq h(\sqrt{n}) + 2$ <p>which solves to $$h(n)= O(\log\log n)$$.</p> <p><strong>Analyzing the treewidth $$\mathrm{tw}(K)$$.</strong> Note that the treewidth of $$\mathcal T_B$$ and all $${\mathcal{T_i}}$$ (before adding boundary vertices in line 9) is bounded by $$\mathrm{tw}(\sqrt{n})$$. Line 9 increases the treewidth of $${\mathcal{T_i}}$$ by at most $$2$$. Since the treewidth of $$K$$ is the maximum treewidth $$\mathcal T_B$$ and all $${\mathcal{T_i}}$$, we have:</p> $\mathrm{tw}(n) \leq \mathrm{tw}(\sqrt{n}) + 2$ <p>which solves to $$\mathrm{tw}(n)= O(\log\log n)$$.</p> <p>This completes the proof of Theorem 1 for the path graph $$P_n$$.</p> <h1 id="2-trees">2. Trees</h1> <p>What is needed to extend the construction of a path graph to a general tree? Two properties we exploited in the construction of $$P_n$$:</p> <ol> <li>$$P_n$$ can be decomposed into $$\sqrt{n}$$ (connected) subpaths.</li> <li>Each subpath in the decomposition has at most two boundary vertices.</li> </ol> <p>Property 2 implies that the total number of boundary vertices is about $$\sqrt{n}$$, which plays a key role in analyzing the top level recursion.</p> <p>It is well known that we can obtain a somewhat similar but weaker decomposition for trees: one can decompose a tree of $$n$$ vertices to roughly $$\sqrt{n}$$ connected subtrees such that the number of boundary vertices is $$\sqrt{n}$$. (A vertex is a boundary vertex of a subtree if it is incident to an edge not in the subtree.) This decomposition is weaker in the sense that a subtree could have more than 2, and indeed up to $$\Omega(\sqrt{n})$$, boundary vertices. Is this enough?</p> <p>Not quite. To glue the tree decomposition $$\mathcal T_i$$ to $$\mathcal{T}$$ (and effectively to $$\mathcal T_B$$), we rely on the fact that there is a bag $$X\in \mathcal T_B$$ containing both boundary vertices in line 11. The analogy for trees would be: there exists a bag $$X$$ containing all boundary vertices of each subtree. This is problematic if a subtree has $$\Omega(\sqrt{n})$$ boundary vertices.</p> <p>Then how abound guaranteeing that each subtree has $$O(1)$$ vertices, say 3 vertices. Will this be enough? The answer is pathetically no. To guarantee 3 vertices in the same bag, one has to add a clique of size 3 between the boundary vertices in the top level recursion. What it means is that, the graph between boundary vertices on which we recursively call the shortcuting procedure is <em>no longer a tree</em>. Thus, we really need a decomposition where every subtree has <strong>at most 2 boundary vertices</strong>.</p> <hr /> <p><strong>Lemma 1:</strong> Let $$T$$ be any tree of $$n$$ vertices, one can decompose $$T$$ into a collection $$\mathcal{D}$$ of $$O(\sqrt{n})$$ subtrees such that every tree $$T’\in \mathcal{D}$$ has $$\lvert V(T’)\rvert \leq \sqrt{n}$$ and at most 2 boundary vertices.</p> <hr /> <p>Lemma 1 is all we need to prove Theorem 1, following exactly the same construction for the path graph $$P_n$$; the details are left to readers.</p> <blockquote> <p><strong>Remark 3:</strong> In developing our shortcutting tree result, we were unaware that Lemma 1 was already known in the literature. A reviewer later pointed out that one can get Lemma 1 from a weaker decomposition using <em>least common ancestor closure</em> , which I reproduce below.</p> </blockquote> <p><strong>Proof of Lemma 1:</strong> First, decompose $$T$$ into a collection $$\mathcal{D}’$$ of $$O(\sqrt{n})$$ subtrees such that each tree in $$\mathcal{D}’$$ has size at most $$\sqrt{n}$$ and that the total number of boundary vertices is $$O(\sqrt{n})$$. As mentioned above, this decomposition is well known; see Claim 1 in our paper  for a proof.</p> <p>Let $$A_1$$ be the set of boundary vertices; $$\lvert A_1\rvert = O(\sqrt{n})$$. Root $$T$$ at an arbitrary vertex. Let $$A_2$$ be the set containing the ancestor of every pair of vertices in $$A_1$$. Let $$B = A_1\cup A_2$$.</p> <p>It is not hard to see that $$\lvert A_2\rvert \leq \lvert A_1\rvert - 1 = O(\sqrt{n})$$. Thus, $$\lvert B\rvert = O(\sqrt{n})$$. Furthermore, every connected component of $$T\setminus B$$ has at most $$\sqrt{n}$$ vertices and has edges to at most 2 vertices in $$B$$. The set $$B$$ induces a decomposition $$\mathcal{D}$$ of $$T$$ claimed in Lemma 1.</p> <h1 id="3-conclusion">3. Conclusion</h1> <p>The emulator in Theorem 1 can be constructed in time $$O(n \log \log n)$$; see Theorem 10 our paper  for more details. The major open problem is:</p> <p><strong>Open problem:</strong> Is $$O((\log \log n)^2)$$ the best possible bound for the product $$\mathrm{tw}(K)\cdot h(K)$$?</p> <p>This open problem is intimately connected to another problem: embedding planar graphs of diameter $$D$$ into low treewidth graphs with additive distortion at most $$+\epsilon D$$ for any $$\epsilon \in (0,1)$$. More precisely, though not explicitly stated , one of our main results is:</p> <hr /> <p><strong>Theorem 2:</strong> If one can construct an emulator $$K$$ of treewidth $$\mathrm{tw}(n)$$ and hop bound $$h(n)$$ for any tree of $$n$$ vertices, then one can embed any planar graphs with $$n$$ vertices and diameter $$D$$ into a graph of treewidth $$O(h(n)\cdot \mathrm{tw}(n)/\epsilon)$$ and additive distortion $$+\epsilon D$$ for any given $$\epsilon \in (0,1)$$.</p> <hr /> <p>That is, the product of treewdith and hop bound directly bounds the treewdith of the embedding. Theorem 1 give us an embedding with treewidth $$O((\log\log n)^2/\epsilon)$$, which has various algorithmic applications . My belief is that the bound $$O((\log \log n)^2)$$ is the best possible.</p> <h1 id="4-references">4. References</h1> <p> Bodlaender, H.L. and Hagerup, T. (1995). Parallel algorithms with optimal speedup for bounded treewidth. In ICALP ‘95, 268-279.</p> <p> Filtser, A. and Hung, L. (2022). Low Treewidth Embeddings of Planar and Minor-Free Metrics. ArXiv preprint <a href="https://arxiv.org/abs/2203.15627">arXiv:2203.15627</a>.</p> <p> Fomin, F.V., Lokshtanov, D., Saurabh, S. and Zehavi, M. (2019). Kernelization: theory of parameterized preprocessing. Cambridge University Press.</p>Hung LeOver the past two years or so, I have been thinking about a cute problem, which turned out to be much more useful to my research than I initially thought. Here it is: Tree Shortcutting Problem: Given an edge-weighted tree $$T$$, add (weighted) edges to $$T$$, called shortcuts, to get a graph $$K$$ such that: $$d_K(u,v) = d_T(u,v)~\quad \forall u,v\in V(T)$$. That is, $$K$$ preserves distances in $$T$$. For every $$u,v\in V(T)$$, there exists a shortest path from $$u$$ to $$v$$ in $$K$$ containing at most $$k$$ edges for some $$k\geq 2$$. The goal is to minimize the product $$k \cdot \mathrm{tw}(K)$$, where $$\mathrm{tw}(K)$$ is the treewidth of $$K$$.Sparsity of minor-free graphs2022-11-30T00:00:00+00:002022-11-30T00:00:00+00:00https://minorfree.github.io/minor-sparsity<p>Planar graphs are sparse: any planar graph with $$n$$ vertices has at most $$3n-6$$ edges. A simple corollary of this sparsity is that planar graphs are $$6$$-colorable. There is simple and beautiful proof based on the Euler formula, which can easily be exteded to bounded genus graphs, a more general case: any graph embedddable in orientable surfaces of genus $$g$$ with $$n$$ vertices has at most $$3n + 6g-6$$ edges.</p> <p>How’s about the number of edges of $$K_r$$-minor-free graphs? This is a very challenging question. A reasonable speculation is $$O(r)\cdot n$$: a disjoint union of $$n/(r-1)$$ copies of $$K_{r-1}$$ excludes a $$K_r$$ minor and has $$\Theta(r)\cdot n$$ edges. But this isn’t the case. And surprisingly, the correct bound is $$O(r\sqrt{\log r})n$$, which will be the topic of this post.</p> <hr /> <p><strong>Theorem 1:</strong> Any $$K_r$$-minor-free graphs with $$n$$ vertices has at most $$O(r\sqrt{\log r})\cdot n$$ edges.</p> <hr /> <p>The bound in Theorem 1 is tight; see a lower bound in Section 4. The sparsity bound, which is the ratio of the number of edges to the number of vertices, $$O(r\sqrt{\log r})$$ was first discovered by Kostochka ; the proof is quite non-trivial, so as other follow-up proofs. A short proof was just found recently by Alon, Krivelevich, and Sudakov (AKS) , which I will present here in Section 3. See the bibliographical notes section for a detailed discussion of other proofs.</p> <p>The goal of this post isn’t just to present the proof of Theorem 1. At various points in the past, I am interested in a more intuitive proof that gives good enough sparsity bounds, say $$O(\mathrm{poly}(r))$$, or even $$O(f(r))$$ bound for any function that depends on $$r$$ only. This post describes different proofs, of increasing complexity (or ingenuity), giving different bounds. In particular, sparsity bound $2^{r}$ follows by a simple induction. Sparsity bound $$O(r^2)$$ relies crucially on the fact that a graph of sparsity $$d$$ has a highly connected minor of small size; the high connectivity allows us to show that for any vertex subset of size $$k \approx \sqrt{d}$$ has $${k \choose 2}$$ internally vertex-disjoint paths connecting these vertices, thereby giving us a $$K_{\sqrt{d}}$$ minor. The optimlal bound $$O(r\sqrt{\log r})$$ also relies on a highly connected minor of small size, but employs a clever probabilistic argument to construct a $$K_{r}$$ minor.</p> <h1 id="1-exponential-sparsity-2r">1. Exponential Sparsity: $$2^{r}$$</h1> <p>I learn a beatiful proof of the following theorem in the graph theory book of Reinhard Diestel (Proposition 7.2.1. ).</p> <hr /> <p><strong>Theorem 2:</strong> Any $$K_r$$-minor-free graphs with $$n$$ vertices has at most $$2^{r-1}\cdot n$$ edges.</p> <hr /> <p>Proof: Let $$G$$ be a $$K_r$$-minor-free graph with $$n$$ vertices. The proof is by induction on $$\vert V(G) \vert + r$$. If there is a vertex $$v$$ of degree at most $$2^{r-1}$$, then by removing $$v$$ and applying the induction hypothesis, we are done. Now consider the case where every vertex has degree more than $$2^{r-1}$$; let $$v$$ be such a vertex. The key idea is to find an edge $$(u,v)$$ such that $$u$$ and $$v$$ share only a few neighbors. We then contract $$(u,v)$$ and apply induction.</p> <blockquote> <p>Claim 1: There is a neighbor $$u$$ of $$v$$ such that $$\vert N_G(v)\cap N_G(u) \vert \leq 2^{r-1}-1$$.</p> </blockquote> <p>Suppose that the claim holds, then let $$G’$$ be the graph obtained from $$G$$ by contracting $$(u,v)$$, i.e., $$G’= G/(u,v)$$. Then $$\vert V(G’) \vert \leq r-1$$ and $$\vert E(G’) \vert \geq \vert E(G) \vert - 2^{r-1}$$. By induction, $$\vert E(G’) \vert \leq 2^{r-1}(n-1)$$, which implies $$\vert E(G) \vert \leq 2^{r-1}(n-1) + 2^{r-1} = 2^{r-1}\cdot n$$ as desired.</p> <p>We now turn to Claim 1. Let $$H = G[N_G(v)]$$ be the sugraph induced by $$N_G(v)$$. Then $$H$$ is $$K_{r-1}$$-minor-free. By induction, $$\vert E(H) \vert \leq 2^{r-2}\cdot \vert V(H) \vert$$. Thus, there exists a vertex $$u\in H$$ such that $$d_H(u) \leq 2 \vert E(H) \vert / \vert V(H) \vert \leq 2^{r-1}$$. This gives $$\vert N_G(v)\cap N_G(u) \vert \leq d_H(u)-1 = 2^{r-1}-1$$.</p> <hr /> <p>The exponential term $$2^{r-1}$$ in the above theorem is due to a loss of a factor of $$2$$ in each step of the induction by using $$d_H(v) \leq 2( \vert E(H) \vert / \vert V(H) \vert )$$.</p> <h1 id="2-polynomial-sparsity-or2">2. Polynomial Sparsity: $$O(r^2)$$</h1> <p>The goal of this section is to improve the exponential bound in Theorem 2 to a polynomial bound.</p> <hr /> <p><strong>Theorem 3:</strong> Any $$K_r$$-minor-free graphs with $$n$$ vertices has at most $$O(r^2)\cdot n$$ edges.</p> <hr /> <p>Proving Theorem 3 requires a deeper understanding of the structures of minor-free graphs. The key insight is that any graph with at least $$d\cdot n$$ edges has a minor, say $$K$$, of $$O(d)$$ size that is highly connected (Lemmas 1 and 2 below). Then we then show that given any set of vertices, say $$R$$, of size about $$\Theta(\sqrt{d})$$ in $$K$$, we can find a collection of pairwise disjoint paths of $$H$$ connecting every pair of vertices in $$R$$ (due to high connectivity), which gives us a clique minor of size $$\Theta(\sqrt{d})$$ (Lemma 3). Taking contrapositive gives Theorem 3.</p> <p>Let $$\varepsilon(G) = \vert E(G) \vert / \vert V(G) \vert$$ and $$\delta(G)$$ be the minimum degree of $$G$$. Fist, we show that $$G$$ contains a minor of size $$\leq 2d$$ and minimum degree at least $$d$$.</p> <hr /> <p><strong>Lemma 1:</strong> Let $$G$$ be any graph of $$n$$ vertices such that $$\vert E(G) \vert \geq d\cdot n$$. Then $$G$$ has a minor $$H$$ such that $$\delta(H)\geq d \geq \vert V(H) \vert /2$$.</p> <hr /> <p>Proof: Let $$K$$ be a <em>minimal</em> minor of $$G$$ such that $$\varepsilon(K)\geq d$$. The minimality implies two properties:</p> <ol> <li>$$\vert N_K(u)\cap N_K(v) \vert \geq d$$ for every edge $$(u,v)$$; otherwise, we can contract edge $$(u,v)$$.</li> <li>There exists a vertex $$x$$ such that $$d_K(x)\leq 2d$$; otherwise, we can delete an edge from $$K$$.</li> </ol> <p>Then $$H = K[N_K(x)]$$ satisfying the lemma.</p> <hr /> <p>Next, we show that $$H$$ has a highly connected subgraph.</p> <hr /> <p><strong>Lemma 2:</strong> Let $$H$$ be such that $$\delta(H)\geq d \geq \vert V(H) \vert /2$$. Then $$H$$ has a subgraph $$K$$ that has (i) $$\vert V(K) \vert \leq 2d$$ and (ii) $$K$$ is $$d/3$$-vertex connected.</p> <hr /> <p>Proof: If $$H$$ is $$d/3$$-vertex connected, then $$K = H$$. Otherwise, there is a set $$S\subseteq V(H)$$ of size at most $$d/3$$ such that $$H\setminus S$$ has at least two connected components. Let $$K$$ be the smallest connected component of $$H$$. Then $$\vert V(K) \vert \leq \vert V(H) \vert /2\leq d$$ and $$\delta(K)\geq \delta(H) - \vert S \vert \geq 2d/3$$. Thus, for every $$u\not= v\in K$$, $$u$$ and $$v$$ must share at least $$\vert N_K(u) \vert + \vert N_K(v) \vert - \vert V(K) \vert \geq d/3$$ neighbors in $$K$$. That is, $$K$$ is $$d/3$$-vertex connected.</p> <hr /> <p><strong>A detour: diameter of highly connected graphs.</strong> Graph $$K$$ in Lemma 2 has another nice property: its diameter is at most $$7$$. To see this, suppose that there is a shortest path $${v_0,v_1,\ldots,v_8, \ldots}$$ of length at least $$8$$. Consider three vertices $$v_1,v_4, v_7$$. Then $$N_{K}(v_1),N_{K}(v_4), N_{K}(v_7)$$ are pairwise disjoint. As $$K$$ is $$d/3$$-connected, $$\delta(K)\geq 3$$. Thus, $$\vert N_{K}(v_1)\cup N_{K}(v_4) \cup N_{K}(v_7) \vert \geq 3(d/3+1) &gt; d\geq V(K)$$, a contradiction. We will use the same kind of arguments in several proofs below.</p> <p>We now construct a minor of size $$\Theta(\sqrt{d})$$ for graph $$K$$ in Lemma 2. We do so by showing that for any given $$p \leq d/40$$ distinct pairs of vertices $${(s_1,t_1, \ldots, (s_p,t_p)}$$ in $$K$$ (two pairs might share the same vertex), then there are $$p$$ internally vertex-disjoint paths connecting them (Lemma 3). (Two paths are internally vertex-disjoint if they can only share endpoints.) Then one can construct a minor of size $$\sqrt{d/40}$$ by picking an arbitrary set $$R$$ of $$\sqrt{d/40}$$ vertices, and connect all pairs of vertices in $$R$$ using disjoint paths in Lemma 3, which implies Theorem 3.</p> <p><img src="/assets/figs/paths-large.svg" alt="" /></p> <p><em>Figure 1: (a) $$\mathcal{P}$$ includes two paths of black edges. (b) deleting $$\mathcal{P}$$ except $$s_1,t_1$$. (c) $$v$$ could not have more than 3 neighbors on the path from $$s_i$$ to $$t_i$$</em></p> <hr /> <p><strong>Lemma 3:</strong> Let $$\mathcal{T} = {(s_1,t_1), \ldots, (s_p,t_p)}$$ be any $$p \leq d/40$$ distinct pairs of vertices in $$K$$ (in Lemma 2). Then there are $$p$$ internally vertex-disjoint paths connecting the all pairs in $$\mathcal{T}$$.</p> <hr /> <p>Proof: Let $$\mathcal{P}$$ be a set of internally vertex-disjoint paths, each of of length at most $$10$$, that connects a maximal number of pairs in $$\mathcal{T}$$. Subject to the pairs connected by $$\mathcal{P}$$, we choose $$\mathcal{P}$$ such that the total number of edges of paths in $$\mathcal{P}$$ is minimal. If $$\mathcal{P}$$ connects every pair, we are done. Otherwise, w.l.o.g, we assume that $$s_1$$ and $$t_1$$ are not connected by $$\mathcal{P}$$. See Figure 1(a).</p> <p>Let $$K^-$$ be obtained from $$K$$ be removing all vertices in $$\mathcal{P}$$, except $$s_1$$ and $$t_1$$, from $$K$$. See Figure 1(b). Observe that the total number of vertices in $$\mathcal{P}$$ is at most $$11\cdot p = 11d/40 &lt; d/3$$. Since $$K$$ is $$d/3$$-vertex-connected, $$K^-$$ is connected.</p> <blockquote> <p>Claim 2: for every $$v\in K^-$$ and $$P\in \mathcal{P}$$, $$\vert N_K(v)\cap V(P) \vert \leq 3$$.</p> </blockquote> <p>Suppose the claim is not true, then we can shortcut $$P$$ via $$v$$ to get a shorter path connecting the endpoints of $$P$$, contradicting the minimality of $$\mathcal{P}$$. See Figure 1(c).</p> <p>Note that $$\delta(K)\geq d/3$$ as it is $$d/3$$-connected. By Claim 2, $$\delta(K^-)\geq \delta(K) -3\cdot p \geq d/3 - 3d/30 \geq d/4$$. The same argument in the detour above implies that $$K^-$$ has diameter at most $$10$$. Thus, there is a path of length at most $$10$$ from $$s_1$$ to $$t_1$$ in $$K^-$$, contradicting the maximality of $$\mathcal{P}$$.</p> <hr /> <h1 id="3-optimal-sparsity-orsqrtlogr">3. Optimal Sparsity: $$O(r\sqrt{\log{r}})$$</h1> <p>We assume that the graph has at least $$d\cdot n$$ edges. Our goal is to construct a minor of size $$\Omega(d/\sqrt{\log d})$$. By taking contrapositive, we obtain Theorem 1.</p> <h2 id="31-proof-ideas">3.1. Proof Ideas</h2> <p>By Lemma 2, it suffices to construct a clique minor of size $$\Omega(d/\sqrt{\log d})$$ for a $$d/3$$-vertex-connected graph $$K$$ with at most $$2d$$ vertices. In particular, we will construct a collection of $$h = d/(c_1 \sqrt{\log d})$$ vertex-disjoint connected subgraphs $$C_1,C_2,\ldots, C_h$$ such that (a) there is an edge between any two subgraphs $$C_i,C_j$$ for $$1\leq i\not=j \leq h$$ and (b) each $$C_i$$ has $$O(c_0) \sqrt{\log d}$$ vertices, for some constant $$1\ll c_0 \ll c_1$$. These subgraphs will realize a $$K_h$$-minor of $$K$$.</p> <p>The choices of constants $$c_0$$ and $$c_1$$ imply that $$\vert V(C_1)\cup \ldots \cup V(C_h) \vert \leq d/12$$. Thus, if we let $$H_i = K\setminus (C_1\cup \ldots \cup C_i)$$ for any $$i\leq h$$, then $$H_i$$ is $$d/3 - d/12 = d/4$$ vertex-connected. This in particular, implies that $$H_i$$ has diameter at most $$22 = O(1)$$.</p> <p><img src="/assets/figs/minor-large.png" alt="" /></p> <p><em>Figure 2: (a) $$C_1$$ forms from $$S_1$$, the set of black vertices, and its bad set $$B_1$$. (b) $$C_2$$ is constructed from $$S_2$$ (black vertices), which avoids $$B-1$$, and its bad set $$B_2$$. (c) $$S_i$$ has edges to all graphs $$C_1,C_2,\ldots,C_{i-1}$$.</em></p> <p>We will construct each $$C_i$$ by random sampling. To gain intuition, let’s look at the first step: (1) sampling a set $$S_1$$ of $$s = c_0\sqrt{\log n}$$ vertices and (2) making $$S_1$$ connected by adding a shortest path from one vertex to every other vertex in $$S_1$$. See Figure 2 (a). There are two good reasons for doing this:</p> <ol> <li>As the graph has diameter $$O(1)$$, $$\vert V(C_1) \vert = O(c_0 \sqrt{\log d})$$.</li> <li>About $$e^{-O(c_0)\sqrt{\log d}}\cdot d$$ vertices are <strong>not dominated</strong> by $$S_1$$. This is because each vertex has at least $$d/3$$ neighbors (as $$K$$ is $$d/3$$-connected), and hence probability that a vertex is not dominated by $$S_1$$ is at most $$(1-d/(3\cdot 2d))^{s} = e^{-O(c_0)\sqrt{\log d}}$$. Let’s call these vertices bad vertices (for a reason explained later), and denote the set of bad vertices by $$B_1$$. (A vertex is donimated by $$S_1$$ if it is in $$S_1$$ or adajcent to another vertex in $$S_1$$.)</li> </ol> <p>The second step, we sample $$S_2$$ in exactly the same way and make $$C_2$$ by adding paths between vertices of $$S_2$$. The graph we construct $$C_2$$ now is $$H_1 = K\setminus V(C_1)$$, and as argue above, $$H_1$$ has roughly the same properties of $$K$$: $$d/4$$-vertex-connected and diameter $$O(1)$$. We want $$S_2$$ to contain a vertex adjacent to $$S_1$$ (in $$K$$) because we would like $$C_2$$ to be adjacent to $$C_1$$. That is, we want $$S_2 \not\subseteq B_1$$: we say that $$S_2$$ <strong>avoids</strong> bad set $$B_1$$. See Figure 2(b). The reason 2 above implies that $$\mathrm{Pr}[S_2\subseteq B_1] \leq (e^{-O(c_0)\sqrt{\log d}})^{c_0\sqrt{\log d}} \leq 1/d^2$$ for some chocie of $$c_0\gg 1$$. Thus, w.h.p, $$S_2$$ avoids $$B_1$$.</p> <p>In general, at any step $$i \in [1,h]$$, we already constructed a set of $$i-1$$ vertex-disjoint conected subgraphs $$C_1,C_2,\ldots C_{i-1}$$, each is associated with a bad set (a set of non-neighbors). See Figure 2(c). We want to construct $$C_i$$ by sampling a set $$S_i$$ and adding paths between vertices of $$S_i$$. By the same reasoning above for $$S_2$$ and using the union bound, the probability that $$S_i$$ is connected to all $$i-1$$ subgraphs, i.e, $$S_i$$ avoids all the $$(i-1)$$ bad sets, is at least $$1 - d/d^2 = 1-1/d &gt; 0$$. When $$i = h$$, we obtain a $$K_h$$ minor as desired.</p> <h2 id="32-the-formal-proof">3.2. The formal proof</h2> <p>Notation: for a given set $$S\subseteq V$$ in a graph $$G = (V,E)$$, denoted by $$B_G(S)$$ be the set of vertices not dominated by $$S$$. That is, $$B_G(S) = V\setminus (S\cup(\cup_{v\in S}N_G(v)))$$.</p> <p>We construct a set of subgraphs $$C_1,C_2,\ldots, C_h$$ realizing a $$K_h$$-minor of $$K$$, for $$h = d/(c_1 \sqrt{\log d})$$, in $$h$$ iterations as follows.</p> <hr /> <p>Initially, $$H_0 = K, B_0 = \emptyset$$.</p> <blockquote> <p>In $$i$$-th iteration, $$i\geq 1$$, we find a set $$S_i$$ of at most $$c_0\sqrt{\log d}$$ vertices s.t (a) $$\vert B_{H_{i-1}}(S_i) \vert \leq 2de^{-c_0\sqrt{\log d}/8}$$ and (b) $$S_i$$ is connected to each of $$C_1,C_2,\ldots,C_{i-1}$$ by an edge. Next, let $$C_i$$ be obtained by adding shortest paths from an arbitrary vertex $$v\in S_i$$ to every other vertex in $$S_i\setminus {v}$$, and $$B_i= B_{H_{i-1}}(S_i)$$. Then, we define $$H_i = H_{i-1}\setminus V(C_i)$$ for the next iteration.</p> </blockquote> <p>Finally, output $$C_1,C_2,\ldots, C_h$$.</p> <hr /> <p>To show the correctness of the algorithm, we only have to show that the set $$S_i$$ at iteration $$i$$ exists, for some choices of $$1\ll c_0 \ll c_1$$. If so, $$C_1,C_2,\ldots, C_h$$ form a $$K_h$$-minor of $$K$$, and hence, of $$G$$.</p> <p>First, we show that $$H_i$$ has high connectivity and $$C_i$$ has size $$O(c_0\sqrt{\log d})$$.</p> <hr /> <p><strong>Lemma 4:</strong> For every $$i\geq 1$$, $$\vert V(C_i) \vert \leq 22 c_0\sqrt{\log d}$$ and $$H_i$$ is $$d/4$$-vertex connected when $$c_1 = 12c_0$$.</p> <hr /> <p>Proof: We prove by induction. Since $$H_{i-1}$$ is $$d/4$$-connected and $$\vert V(H_i) \vert \leq 2d$$, the diameter of $$H_{i-1}$$ is at most $$22$$. As we add at most $$c_0\sqrt{\log d}$$ shortest paths to $$S_i$$, $$\vert V(C_i) \vert \leq 22 c_0\sqrt{\log d}$$.</p> <p>Observe that $$\sum_{j=1}^{i} \vert V(C_j) \vert \leq c_0\sqrt{\log d}\cdot h= c_0\sqrt{\log d} \cdot d/(c_1\sqrt{\log d}) = d/12$$. Since $$K$$ is $$d/3$$-vertex connected, $$H_i$$ is $$d/3-d/12 = d/4$$ vertex connected.</p> <hr /> <p>Now we show the existence of $$S_i$$. Note that condition (b) is equivalent to that $$S_i$$ avoids all the bad sets $$B_0,B_1,\ldots, B_{i-1}$$. Let $$S_i$$ be otabined by choosing each vertex of $$H_{i-1}$$ with probability $$c_0\sqrt{\log d}/(2d)$$; the expected size of $$S_i$$ is a most $$c_0\sqrt{\log d}$$. By Lemma 4, every vertex $$v\in H_{i-1}$$ has degree at least $$d/4$$. Thus, $$\mathrm{Pr}[v\in B_{i}]\leq (1-c_0\sqrt{\log d}/(2d))^{d/4}\sim e^{-c_0\sqrt{\log d}/8}$$. In particular, $$\vert \mathbb{E}[B_i] \vert \leq (2d)e^{-c_0\sqrt{\log d}/8}$$.</p> <p>It remains to show that with non-zero probability, $$S_i$$ avoids all $$B_0,\ldots, B_{i-1}$$. Note that $$\vert V(H_{i-1}) \vert \geq d/4$$. For a fixed $$j\in [0,i-1]$$:</p> <p>$$\mathrm{Pr}[S_i\subseteq B_j]\leq ( \vert B_j \vert / \vert V(H_{i-1}) \vert )^{ \vert S_j \vert }\leq 8(e^{-c_0\sqrt{\log d}/8})^{c_0\sqrt{\log d}}= 8e^{-c_0^2 \log(d)/8} \leq 1/d^2$$</p> <p>for a sufficiently large $$c_0\geq 1$$.</p> <p>By union bound, the probability that $$\mathrm{Pr}[S_i\subseteq B_j]$$ for some $$j\in [0,i-1]$$ is at most $$h/d^2\leq 1/d$$. Thus, the probability that $$S_i$$ avoids all $$B_j$$ is at least $$1-1/d$$. This conclude the proof.</p> <h1 id="4-a-lower-bound">4. A Lower Bound</h1> <p>In this section, we show that for any $$n$$ and $$r$$ such that $$n \gg r\sqrt{\log r}$$, there exists a graph $$G$$ with $$n$$ vertices and $$\Theta(n\cdot r\sqrt{\log r})$$ edges such that $$G$$ has no $$K_{r}$$ minor. The key idea of the construction is Theorem 4 below.</p> <hr /> <p><strong>Theorem 4:</strong> There exists a graph $$H$$ with $$k$$ vertices and $$\Theta(k^2)$$ edges such that $$H$$ has no $$K_s$$-minor where $$s = k/(\epsilon\sqrt{\log k})$$ for some constant $$\epsilon\in (0,1)$$.</p> <hr /> <p>Theorem 4 implies a sparsity lower bound $$\Omega(r\sqrt{\log r})$$ as follows. Let $$G$$ be the disjoint union of $$\Theta(n/(r\sqrt{\log r}))$$ copies of the same graph in Theorem 4 with $$k = \Theta(r\sqrt{\log r})$$ vertices. Then $$\vert E(G) \vert = \Theta(n/(r\sqrt{\log r}))k^2 = \Theta(n\cdot r\sqrt{\log r})$$. As $$H$$ excludes a clique minor of size $$k/(\epsilon\sqrt{\log k}) \leq r$$ (by choosing the constant in the definition of $$k$$ appropriately), $$G$$ excludes $$K_r$$ as a minor.</p> <p>Theorem 4 can be proven by the probabilistic method. To gain some intuition of the proof, consider any fixed partition of $$V(H)$$ into vertex-disjont subsets $${V_1,V_2,\ldots, V_{s}}$$ of size $$\epsilon \sqrt{\log k}$$ each. For this partition to realize a $$K_s$$ minor, there must be an edge between every two vertex sets $$V_i,V_j$$ for $$i\not=j$$. The probability of this is:</p> $(1-2^{-|V_i||V_j|})^ = (1-2^{-\epsilon ^2 \log(k)})^ \approx e^{-k^{2 - \epsilon^2}/\log(k)}$ <p>By the union bound over at most $$k^k$$ such partitions, the probability of having a $$K_s$$ minor is at most $$k^k e^{-k^{2 - \epsilon^2}/\log(k)} \rightarrow 0$$ when $$k \rightarrow +\infty$$. In other words, the probability of not having a $$K_{s}$$ minor is close to $$1$$.</p> <p>In the formal proof, one has to work with the fact that the subsets in the partition might not have the same size; this can be resolved by simple algebraic manipulation.</p> <p><strong>Proof of Theorem 4.</strong> Let $$H = G(k,1/2)$$ where $$G(k,1/2)$$ is the Erdős–Rényi random graph with probability $$p = 1/2$$. We now show that the probability that $$H$$ contains a $$K_s$$ minor tends to $$0$$ when $$k\rightarrow \infty$$.</p> <p>Recall that $$K_s$$ is a minor of $$H$$ if there is a set of non-empty, connected, and vertex-disjoint subgraphs $$\mathcal{C} = {C_1,C_2,\ldots, C_s}$$ such that there is an edge in $$H$$ connecting every two graphs $$C_i,C_j$$ for $$1\leq i\not=j \leq s$$.</p> <p>We will bound the probability of exsiting such $$\mathcal{C}$$. Observe that the number of (ordered) partitions of $$\vert V(H) \vert$$ into $$s$$ non-empty subset is at most:</p> $\frac{k!}{s!}{k-1\choose s-1} &lt;k^k$ <p>Fixed such a partition of $$\vert V(H) \vert$$, denoted by $$\mathcal{P}$$. Let $$n_i$$ be the number of vertices in $$i$$-th set. The probability that there is an edge betwen two different sets $$i$$ and $$j$$ is $$(1-2^{-n_i\cdot n_j})$$. Thus, probability of having an edge between any two different sets of $$\mathcal{P}$$ is:</p> $\prod_{(i,j)}(1-2^{-n_i\cdot n_j}) \leq \prod_{(i,j)}e^{-2^{-n_i\cdot n_j}} = e^{-\sum_{(i,j)}2^{-n_i\cdot n_j}}$ <p>where the product and sum is over all unordered pairs $$(i,j)$$. This implies that:</p> $\mathrm{Pr}[\mathcal{C} \text{ exists}] \leq k^k \cdot e^{-\sum_{(i,j)}2^{-n_i\cdot n_j}}$ <p>We now estimate $$\sum_{(i,j)}2^{-n_i\cdot n_j}$$. By arithmetic–geometric mean inequality, $$\sum_{(i,j)}2^{-n_i\cdot n_j}\geq {s \choose 2}\left(\prod_{(i,j)}2^{-n_i\cdot n_j}\right)^{1/{s\choose 2}} \geq {s \choose 2} \left(2^{-\sum_{(i,j)}n_i\cdot n_j}\right)^{1/{s\choose 2}}$$</p> <p>Observe that $$\sum_{(i,j)}n_i\cdot n_j$$ is the number of edges in a complete s-partite graph with $$k$$ vertices. Thus, $$\sum_{(i,j)}n_i\cdot n_j\leq k^2/2$$ and hence:</p> $\sum_{(i,j)}2^{-n_i\cdot n_j} \geq {s \choose 2} 2^{-k^2/s^2}$ <p>Thus,</p> $\mathrm{Pr}[\mathcal{C} \text{ exists}] \leq k^k \cdot e^{- {s \choose 2} 2^{-k^2/s^2}}$ <p>By chooosing $$s = ck/(\sqrt{\log k})$$ for some big enough constant $$c$$, we have $$\mathrm{Pr}[\mathcal{C} \text{ exists}] \rightarrow 0$$ when $$k\rightarrow \infty$$.</p> <hr /> <h1 id="bibliographical-notes">Bibliographical Notes</h1> <p>The exponential sparsity bound in Section is due to Reinhard Diestel (Proposition 7.2.1. ). The $$O(r^2)$$ sparsity bound in Section 2 is obtained by a combination of various ideas, in particular, Lemma 1 is due to Exercise 21, Chapter 7, in , Lemma 2 is from the proof of Theorem 1 in , and Lemma 3 is a modification of Lemma 3.5.4 in .</p> <p>Mader  proved a sparstiy bound $$O(r\log(r))$$ for $$K_r$$-minor-free graphs. Kostacha was the first to show that the sparsity is $$\Theta(r\sqrt{\log r})$$. Thomason  provided a more refined range for the sparsity bound: $$[0.265r\sqrt{\log_2 r}(1+o(1)), 0.268r\sqrt{\log_2 r}(1+o(1))]$$. The bound then was tightened <strong>exactly</strong> to $$(\alpha +o(1))r\sqrt{\ln(r)}$$ where $$\alpha = 0.319…$$ is an explcit constant, also by Thomason . The simpler proof presented in Section 3 is due to Alon, Krivelevich, and Sudakov .</p> <p>The lower bound in Section 4 is due to Bollobás, Catlin, and Erdös .</p> <h1 id="references">References</h1> <p> Alon, N., Krivelevich, M., and Sudakov, B. (2022). <em>Complete minors and average degree–a short proof</em>. ArXiv preprint <a href="https://arxiv.org/abs/2202.08530">arXiv:2202.08530</a>.</p> <p> Bollobás, B., Catlin, P. A., and Erdös, P. (1980). <em>Hadwiger’s conjecture is true for almost every graph</em>. Eur. J. Comb., 1(3), 195-199.</p> <p> Diestel, R. (2017). Graph theory. Springer.</p> <p> Kostochka, A. V. (1982). <em>A lower bound for the Hadwiger number of a graph as a function of the average degree of its vertices</em>. Discret. Analyz. Novosibirsk, 38, 37-58.</p> <p> Mader, W. (1968). <em>Homomorphiesätze für graphen</em>. Mathematische Annalen, 178(2), 154-168.</p> <p> Thomason, A. (1984). <em>An extremal function for contractions of graphs</em>. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 95, No. 2, pp. 261-265). Cambridge University Press.</p> <p> Thomason, A. (2001). <em>The extremal function for complete minors</em>. Journal of Combinatorial Theory, Series B, 81(2), 318-338.</p>Hung LePlanar graphs are sparse: any planar graph with $$n$$ vertices has at most $$3n-6$$ edges. A simple corollary of this sparsity is that planar graphs are $$6$$-colorable. There is simple and beautiful proof based on the Euler formula, which can easily be exteded to bounded genus graphs, a more general case: any graph embedddable in orientable surfaces of genus $$g$$ with $$n$$ vertices has at most $$3n + 6g-6$$ edges. How’s about the number of edges of $$K_r$$-minor-free graphs? This is a very challenging question. A reasonable speculation is $$O(r)\cdot n$$: a disjoint union of $$n/(r-1)$$ copies of $$K_{r-1}$$ excludes a $$K_r$$ minor and has $$\Theta(r)\cdot n$$ edges. But this isn’t the case. And surprisingly, the correct bound is $$O(r\sqrt{\log r})n$$, which will be the topic of this post. Theorem 1: Any $$K_r$$-minor-free graphs with $$n$$ vertices has at most $$O(r\sqrt{\log r})\cdot n$$ edges.