A family of graphs $\mathcal{F}$ is hereditary if $\mathcal{F}$ is closed under isomorphism and taking induced subgraphs. The speed of $\mathcal{F}$ is the sequence $\{|\mathcal{F}^n|\}_{n \in \mathbb{N}}$, where $\mathcal{F}^n$ denotes the set of graphs in $\mathcal{F}$ with the vertex set $[n]$. Alon, Balogh, Bollobás and Morris [The structure of almost all graphs in a hereditary property, JCTB 2011] gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family $\mathcal{F}$ there exist $\varepsilon>0$ and an integer $l \geq 1$ such that $$|\mathcal{F}^n| = 2^{(1-1/l)n^2/2+o(n^{2-\varepsilon})}.$$ The main result of this paper gives a more precise description of typical structure for a restricted class of hereditary families. As a consequence we characterize hereditary families with the speed just above the threshold $2^{(1-1/l)n^2/2}$, generalizing a result of Balogh and Butterfield [Excluding induced subgraphs: Critical graphs, RSA 2011].
We give evidence in support of a conjecture that the $\nabla$Y$\nabla$ families of $K_{n+5}$ and $K_{3^2,1^n}$ are obstructions for the $n$-apex property.
Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ of sets are said to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ intersects any set in $\mathcal{A}_j$. For a finite set $X$, let $2^X$ denote the \emph{power set of $X$} (the family of all subsets of $X$). A family $\mathcal{H}$ is said to be \emph{hereditary} if all subsets of any set in $\mathcal{H}$ are in $\mathcal{H}$; so $\mathcal{H}$ is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family $\math…
A graph $G$ is $H$-free if it does not contain an induced subgraph isomorphic to $H$. The study of the typical structure of $H$-free graphs was initiated by Erdős, Kleitman and Rothschild, who have shown that almost all $C_3$-free graphs are bipartite. Since then the typical structure of $H$-free graphs has been determined for several families of graphs $H$, including complete graphs, trees and cycles. Recently, Reed and Scott proposed a conjectural description of the typical structure of $H$-free graphs for all graphs $H$, which extends all previously known results in the area. We construct a…
For a hereditary family of graphs $\FF$, let $\FF_n$ denote the set of all members of $\FF$ on $n$ vertices. The speed of $\FF$ is the function $f(n)=|\FF_n|$. An implicit representation of size $\ell(n)$ for $\FF_n$ is a function assigning a label of $\ell(n)$ bits to each vertex of any given graph $G \in \FF_n$, so that the adjacency between any pair of vertices can be determined by their labels. Bonamy, Esperet, Groenland and Scott proved that the minimum possible size of an implicit representation of $\FF_n$ for any hereditary family $\FF$ with speed $2^{Ω(n^2)}$ is $(1+o(1)) \log_2 |\FF_n…
A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set in $\mathcal{B}$. A family $\mathcal{H}$ is hereditary if for each set $A$ in $\mathcal{H}$, all the subsets of $A$ are in $\mathcal{H}$. The $r$th level of $\mathcal{H}$, denoted by $\mathcal{H}^{(r)}$, is the family of $r$-element sets in $\mathcal{H}$. A set $B$ in $\mathcal{H}$ is a base of $\mathcal{H}$ if for each set $A$ in $\mathcal{H}$, $B$ is n…
A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Families $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_k$ of sets are cross-$t$-intersecting if, for every $i$ and $j$ in $\{1, 2, \dots, k\}$ with $i \neq j$, each set in $\mathcal{A}_i$ $t$-intersects each set in $\mathcal{A}_j$. An active problem in extremal set theory is to determine, for a given finite family $\mathcal{F}$, the structure of $k$ cross-$t$-intersecting subfamilies whose sum or product of sizes is maximum. For a family $\mathcal{H}$, the $r$-th level $\mathcal{H}^{(r)}$ of $\mathcal{H}$ …
A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if any two sets in $\mathcal{A}$ have at least $t$ common elements. A central problem in extremal set theory is to determine the size or structure of a largest $t$-intersecting sub-family of a given family $\mathcal{F}$. We give a survey of known results, conjectures and open problems for various important families $\mathcal{F}$, namely, power sets, levels of power sets, hereditary families, families of signed sets, families of labeled sets, and families of permutations. We also provide some extensions and consequences of kno…
It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph fam…
We introduce the notion of an M-family of infinite subsets of $\nn$ which is implicitly contained in the work of A. R. D. Mathias. We study the structure of a pair of orthogonal hereditary families $\aaa$ and $\bbb$, where $\aaa$ is analytic and $\bbb$ is $C$-measurable and an M-family.
We prove that Aldous' Brownian CRT is the scaling limit, with respect to the Gromov--Prokhorov topology, of uniform random graphs in each of the three following families of graphs: distance-hereditary graphs, $2$-connected distance-hereditary graphs and $3$-leaf power graphs. Our approach is based on the split decomposition and on analytic combinatorics.
The implicit representation conjecture concerns hereditary families of graphs. Given a graph in such a family, we want to assign some string of bits to each vertex in such a way that we can recover the information about whether 2 vertices are connected or not using only the 2 strings of bits associated with those two vertices. We then want to minimise the length of this string. The conjecture states that if the family is hereditary and small enough (it only has $2^{O(n\ln(n))}$ graphs of size $n$), then $O(\ln(n))$ bits per vertex should be sufficient. The trivial bounds on this problem are th…
For a fixed graph $H$ and for arbitrarily large host graphs $G$, the number of homomorphisms from $H$ to $G$ and the number of subgraphs isomorphic to $H$ contained in $G$ have been extensively studied in extremal graph theory and graph limits theory when the host graphs are allowed to be dense. This paper addresses the case when the host graphs are robustly sparse and proves a general theorem that solves a number of open questions proposed since 1990s and strengthens a number of results in the literature. We prove that for any graph $H$ and any set ${\mathcal H}$ of homomorphisms from $H$ to …
Minimal obstructions to $C_5$-coloring in hereditary graph classes
KU Leuven Campus Kulak-Kortrijk（荷语鲁汶大学库拉-科特赖克校区）；Ghent University（根特大学）；Warsaw University of Technology（华沙理工大学）；University of Warsaw（华沙大学）；University of Cologne（科隆大学）
Let $\ee>0$ and $\fff$ be a family of finite subsets of the Cantor set $\ccc$. Following D. H. Fremlin, we say that $\fff$ is $\ee$-filling over $\ccc$ if $\fff$ is hereditary and for every $F\subseteq\ccc$ finite there exists $G\subseteq F$ such that $G\in\fff$ and $|G|\geq\ee |F|$. We show that if $\fff$ is $\ee$-filling over $\ccc$ and $C$-measurable in $[\ccc]^{<ω}$, then for every $P\subseteq\ccc$ perfect there exists $Q\subseteq P$ perfect with $[Q]^{<ω}\subseteq\fff$. A similar result for weaker versions of density is also obtained.
An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most $2^{O(n\log(n))}$ graphs on $n$ vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at m…
The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics are the distance functions in graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite graphs. In this paper we prove that the asymptotic dimension of any proper minor-closed family, any class of graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, …
Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs
University of Haifa–Oranim（海法大学奥拉尼姆分校）；First Principles（第一原理）；Rice University（莱斯大学）；University of Johannesburg（约翰内斯堡大学）；University of Houston–Downtown（休斯顿城市大学）
