Related papers: Weak regularity and finitely forcible graph limits
This paper defines, for each graph $G$, a flag vector $fG$. The flag vectors of the graphs on $n$ vertices span a space whose dimension is $p(n)$, the number of partitions on $n$. The analogy with convex polytopes indicates that the linear…
The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
A set $W$ of vertices of $G$ is said to be a weak total resolving set for $G$ if $W$ is a resolving set for $G$ as well as for each $w\in W$, there is at least one element in $W-\{w\}$ that resolves $w$ and $v$ for every $v\in V(G)- W$.…
A connected graph is called fragile if it contains an independent vertex cut. In 2002 Chen and Yu proved that every connected graph of order $n$ and size at most $2n-4$ is fragile, and in 2013 Le and Pfender characterized the non-fragile…
Graphons are symmetric measurable functions that arise from a sequence of graphs. A graphon variety is the a set of all graphons defined by a condition of the form $t(g, W) = 0$ for a fixed quantum graph $g$, where $t(.,.)$ is the…
One of the major results of [N. Robertson and P. D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63(1):65--110, 1995], also known as the weak structure theorem, revealed the local structure of graphs…
Generalizing the concept of dense hypergraph, we say that a hypergraph is weakly dense, if no k in the half-open interval [2,sqrt(n)) is the degree of more than k^2 vertices. In our main result, we prove the famous Erdos-Faber-Lovasz…
Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. Combinatorial optimization studies the…
A set N is called a "weak epsilon-net" (with respect to convex sets) for a finite set X in R^d if N intersects every convex set that contains at least epsilon*|X| points of X. For every fixed d>=2 and every r>=1 we construct sets X in R^d…
The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as $v_1, v_2, \ldots, v_n$, so that for each $2 \le i \le n$, vertex $v_i$ is either adjacent…
A strongly polynomial sequence of graphs $(G_n)$ is a sequence $(G_n)_{n\in\mathbb{N}}$ of finite graphs such that, for every graph $F$, the number of homomorphisms from $F$ to $G_n$ is a fixed polynomial function of $n$ (depending on $F$).…
One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the…
Let G be a finite graph with the non-k-order property (essentially, a uniform finite bound on the size of an induced sub-half-graph). A major result of the paper applies model-theoretic arguments to obtain a stronger version of…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
Local Irregularity Conjecture states that every simple connected graph, except special cacti, can be decomposed into at most three locally irregular graphs, i.e., graphs in which adjacent vertices have different degrees. The connected…
Graphings are special bounded-degree graphs on probability spaces, representing limits of graph sequences that are convergent in a local or local-global sense. We describe a procedure for turning the underlying space into a compact metric…
The function $\Gamma$ on the space of graphons, introduced in [CGH$^+$15], aims to measure the extent to which a graphon $w$ exhibits the Robinson property: for all $x<y<z$, $w(x,z)\leq \min\{ w(x,y),w(y,z)\}$. Robinson graphons form a…
Network complexity has been studied for over half a century and has found a wide range of applications. Many methods have been developed to characterize and estimate the complexity of networks. However, there has been little research with…
Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several well-known eigenvalue bounds. In this work we…