We improve the exposition of the paper and correct our proof for uniform hypergraphs. Additionally, we removed our rederivation of a result of Janson et al. on maximizing homomorphism counts due to an error in the proof of Lemma 4.3
Given a fixed graph $H$ and a constant $c \in [0,1]$, we can ask what graphs $G$ with edge density $c$ asymptotically maximize the homomorphism density of $H$ in $G$. For all $H$ for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any $H$ the maximizing $G$ is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs, having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximization, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs $H$ and densities $c$ such that the optimizing graph $G$ is neither the quasi-star nor the quasi-clique, reproving a result of Day and Sarkar. We also show that for $c$ large enough all graphs $H$ maximize on the quasi-clique, which was also recently proven by Gerbner et al., and for any $c \in [0,1]$ the density of $K_{1,2}$ is always maximized on either the quasi-star or the quasi-clique, which was originally shown by Ahlswede and Katona. Finally, we extend our results to uniform hypergraphs.
Many problems and conjectures in extremal combinatorics concern polynomial inequalities between homomorphism densities of graphs where we allow edges to have real weights. Using the theory of graph limits, we can equivalently evaluate polynomial expressions in homomorphism densities on kernels $W$, i.e., symmetric, bounded, and measurable functions $W$ from $[0,1]^2 \to \mathbb{R}$. In 2011, Hatami and Norin proved a fundamental result that it is undecidable to determine the validity of polynomial inequalities in homomorphism densities for graphons (i.e., the case where the range of $W$ is $[0…
The Kruskal-Katona theorem together with a theorem of Razborov determine the closure of the set of points defined by the homomorphism density of the edge and the triangle in finite graphs. The boundary of this region is a countable union of algebraic curves, and in particular, it is almost everywhere differentiable. One can more generally consider the region defined by the homomorphism densities of a list of given graphs, and ask whether the boundary is as well-behaved as in the case of the triangle and the edge. Towards answering this question in the negative, we construct examples which show…
Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the "middle ranges", the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension $d$, under appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs the "middle" range, and so we expect their stud…
We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization based on a natural system of linear equations whose integer solutions correspond to graph isomorphism. By results of Tinhofer (1986, 1991) and Dvořák (2010), two graphs G and H are fractionally isomorphic if and only if, for every tree T, the number of homomorphisms from T to G equals the corresponding number from T to H, which means that the tree distance of…
The purpose of this article is to show that even the most elementary problems in asymptotic extremal graph theory can be highly non-trivial. We study linear inequalities between graph homomorphism densities. In the language of quantum graphs the validity of such an inequality is equivalent to the positivity of a corresponding quantum graph. Similar to the setting of polynomials, a quantum graph that can be represented as a sum of squares of labeled quantum graphs is necessarily positive. Lovász asks whether the opposite is also true. We answer this question and also a related question of Razbo…
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the quantity $i(G)^{1/v(G)}$, the number of independent sets in $G$ normalized exponentially by the size of $G$? What if $i(G)$ is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.
Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the general structure of differentiable graphon parameters $F$. We derive consistency conditions among the higher Gâteaux derivatives of $F$ when restricted to the subspace of edge weighted graphs $\mathcal{W}_{\bf p}$. Surprisingly, these constraints are rigid enough to imply that the multilinear functionals $Λ: \mathcal{W}_{\bf p}^n \to \mathbb{R}$ satisfying…
Colored Turán densities and finite directed templates
School of Mathematics, Shandong University（山东大学数学学院）；Universidad Pública de Navarra（纳瓦拉公立大学）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极端组合学与概率组）
Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and edges into small image graphs. For the image graph $H_{\text{ind}}$, the graph on two adjacent vertices, one of which is looped, each homomorphism from $G$ to $H_{\text{ind}}$ corresponds to an independent set in $G$. It follows from the Kruskal-Katona theorem that the number of homomorphisms to $H_{\text{ind}}$ is maximized by the lex graph, whose edges form a…
We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right convergence'' defined in terms of the densities of homomorphisms from the graphs of the sequence into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs, and for graphs with nodeweights and edgeweights. One of the main steps here is th…
A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class $\mathcal{F}$ of graphs, a $k$-edge-colored graph $\mathbb{H}$ (not necessarily with the underlying graph in $\mathcal{F}$) is $k$-universal for $\mathcal{F}$ when any $k$-edge-colored graph with the underlying graph in $\mathcal{F}$ admits a homomorphism to $\mathbb{H}$. We characterize graph …
For graphs $G$ and $H$, what relations can be determined between $t(G,W)$ and $t(H,W)$ for a general graph $W$? We study this problem through the framework of the density domination exponent, which is defined to be the smallest constant $c$ such that $t(G,W)\ge t(H,W)^c$ for every graph $W$. This broad generalization encompasses the Sidorenko conjecture, the Erdős-Simonovits Theorem on paths, and a variety of other statements relating graph homomorphism densities. We introduce some general tools for estimating the density domination exponent, and extend previous results to new graph regimes.
Studying inequalities between subgraph- or homomorphism-densities is an important topic in graph theory. Sums of squares techniques have proven useful in dealing with such questions. Using an approach from real algebraic geometry, we strengthen a Positivstellensatz for simple quantum graphs by Lovász and Szegedy, and we prove several new Positivstellensätze for nonnegativity of quantum multigraphs. We provide new examples and counterexamples.
Let $H$ be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph $G$ without isolated vertices, the weighted number of graph homomorphisms $\hom(G, H)$ satisfies the inequality \[ \hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)}, \] where $d_u$ denotes the degree of vertex $u$ in $G$. In particular, one has \[ \hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2} \] for every $d$-regular triangle-free $G$. The triangle-free hypothesis on $G$ is best possible. More generally, we prove a graphical Brascamp-Lieb type inequalit…
Bipodal optimizers in the upper-tail variational problem for regular subgraph densities
School of Computing, KAIST（KAIST计算机学院）；Korea Institute for Advanced Study (KIAS)（韩国高等科学研究院）；Center for AI and Natural Sciences, Korea Institute for Advanced Study (KIAS)（韩国高等科学研究院人工智能与自然科学中心）；Department of Mathematical Sciences, KAIST（KAIST数学科学系）
We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of $k$-star subgraphs, $k\ge 2$ fixed. We prove that under such constraints graphs are "multipodal": asymptotically in the number of vertices there is a partition of the vertices into $M < \infty$ subsets $V_1, V_2, \ldots, V_M$, and a set of well-defined probabilities $g_{ij}$ of an edge between any $v_i \in V_i$ and $v_j \in V_j$. For $2\le k\le 30$ we determine the phase space: the combinations of edge and $k$-star densities achievable asymptotically. For these models there are special points…
