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 $(1+o(1)) \log_2 |\FF_n|/n~(=Θ(n))$. A recent result of Hatami and Hatami shows that the situation is very different for very sparse hereditary families. They showed that for every $δ>0$ there are hereditary families of graphs with speed $2^{O(n \log n)}$ that do not admit implicit representations of size smaller than $n^{1/2-δ}$. In this note we show that even a mild speed bound ensures an implicit representation of size $O(n^c)$ for some $c<1$. Specifically we prove that for every $\eps>0$ there is an integer $d \geq 1$ so that if $\FF$ is a hereditary family with speed $f(n) \leq 2^{(1/4-\eps)n^2}$ then $\FF_n$ admits an implicit representation of size $O(n^{1-1/d} \log n)$. Moreover, for every integer $d>1$ there is a hereditary family for which this is tight up to the logarithmic factor.
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 …
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…
The idea of implicit representation of graphs was introduced in [S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, SIAM J. Discrete Mathematics, 5 (1992) 596--603] and can be defined as follows. A representation of an $n$-vertex graph $G$ is said to be implicit if it assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency of two vertices is a function of their codes. Since an implicit representation of an $n$-vertex graph uses $O(n\log n)$ bits, any class of graphs admitting such a representation contains $2^{O(n\log n)}$ labelled graphs with $n$…
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…
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…
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 $\mathcal{H}$ …
How to efficiently represent a graph in computer memory is a fundamental data structuring question. In the present paper, we address this question from a combinatorial point of view. A representation of an $n$-vertex graph $G$ is called implicit if it assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency of two vertices is a function of their codes. A necessary condition for a hereditary class $X$ of graphs to admit an implicit representation is that $X$ has at most factorial speed of growth. This condition, however, is not sufficient, as was recently shown in…
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…
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.
Recent works of Alon-Shapira and Rödl-Schacht have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally". In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychr…
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…
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, …
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）
