Related papers: Which subsets of an infinite random graph look ran…

We give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction.…

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…

A \emph{generic rectangular layout} (for short, \emph{layout}) is a subdivision of an axis-aligned rectangle into axis-aligned rectangles, no four of which have a point in common. Such layouts are used in data visualization and in…

We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if…

The \emph{Antimagic Graph Conjecture} asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, ..., |E|$ is used exactly once and the sums of the labels on all edges incident with a…

A graph $H$ is common if its Ramsey multiplicity, i.e., the minimum number of monochromatic copies of $H$ contained in any $2$-edge-coloring of $K_n$, is asymptotically the same as the number of monochromatic copies in the random…

One of the main questions that arise when studying random and quasi-random structures is which properties P are such that any object that satisfies P "behaves" like a truly random one. In the context of graphs, Chung, Graham, and Wilson…

A graph is universally $k$-edge-weightable if for every $k$-element set $Q\subset\mathbb{R}$, it admits a proper $Q$-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression $\{a,b,c\}$, every nice regular…

Causal graph dynamics are transformations over graphs that capture two important symmetries of physics, namely causality and homogeneity. They can be equivalently defined as continuous and translation invariant transformations or functions…

A graph on $n$ vertices is said to be \emph{$C$-Ramsey} if every clique or independent set of the graph has size at most $C \log n$. The only known constructions of Ramsey graphs are probabilistic in nature, and it is generally believed…

Starting from a supercompact cardinal we build a model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$.…

A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present…

Let P_{n,d,D} denote the graph taken uniformly at random from the set of all labelled planar graphs on {1,2,...,n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability…

For a positive integer $n$, a graph with at least $n$ vertices is $n$-existentially closed or simply $n$-e.c. if for any set of vertices $S$ of size $n$ and any set $T\subseteq S$, there is a vertex $x\not\in S$ adjacent to each vertex of…

The distinguishing number of a graph $G$ is the smallest positive integer $r$ such that $G$ has a labeling of its vertices with $r$ labels for which there is no non-trivial automorphism of $G$ preserving these labels. Albertson and Collins…

The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but…

A set $I$ is said to be a universal indifferent set for $1$-genericity if for every $1$-generic $G$ and for all $X \subseteq I$, $G \Delta X$ is also $1$-generic. Miller showed that there is no infinite universal indifferent set for…

We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…

Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has…

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…