arXiv subjects
Publications and source records attributed to Mihyun Kang.
In the planted bisection model a random graph $G(n,p_+,p_- )$ with $n$ vertices is created by partitioning the vertices randomly into two classes of equal size (up to $\pm1$). Any two vertices that belong to the same class are linked by an edge with probability $p_+$ and any two that belong to different classes with probability $p_- c\sqrt{d_+ \ln d_+ }$ for a certain constant $c>0$.
The $k$-core, defined as the largest subgraph of minimum degree $k$, of the random graph $G(n,p)$ has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [JCTB 67 (1996) 111--151] determined the threshold $d_k$ for the appearance of an extensive $k$-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the $k$-core is embedded into the random graph for any $k\geq3$ and any fixed average degree $d=np>d_k$. This generalises prior results on, e.g., the internal structure of the $k$-core.
We consider the following definition of connectivity in $k$-uniform hypergraphs: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We determine the threshold at which the random $k$-uniform hypergraph with edge probability $p$ becomes $j$-connected with high probability. We also deduce a hitting time result for the random hypergraph process -- the hypergraph becomes $j$-connected at exactly the moment when the last isolated $j$-set disappears. This generalises well-known results for graphs.
The stochastic Kronecker graph model introduced by Leskovec et al. is a random graph with vertex set $\mathbb Z_2^n$, where two vertices $u$ and $v$ are connected with probability $α^{{u}\cdot{v}}γ^{(1-{u})\cdot(1-{v})}β^{n-{u}\cdot{v}-(1-{u})\cdot(1-{v})}$ independently of the presence or absence of any other edge, for fixed parameters $0<α,β,γ<1$. They have shown empirically that the degree sequence resembles a power law degree distribution. In this paper we show that the stochastic Kronecker graph a.a.s. does not feature a power law degree distribution for any parameters $0<α,β,γ<1$. In addition, we analyze the number of subgraphs present in the stochastic Kronecker graph and study the typical neighborhood of any given vertex.
The phase transition in the size of the giant component in random graphs is one of the most well-studied phenomena in random graph theory. For hypergraphs, there are many possible generalisations of the notion of a component, and for all but the simplest example, the phase transition phenomenon was first proved by Cooley, Kang and Person. In this paper we build on this and determine the asymptotic size of the unique giant component.
In this paper we consider $j$-tuple-connected components in random $k$-uniform hypergraphs (the $j$-tuple-connectedness relation can be defined by letting two $j$-sets be connected if they lie in a common edge and consider the transitive closure; the case $j=1$ corresponds to the common notion of vertex-connectedness). We determine that the existence of a $j$-tuple-connected component containing $Θ(n^j)$ $j$-sets in random $k$-uniform hypergraphs undergoes a phase transition and show that the threshold occurs at edge probability $\tfrac{(k-j)!}{\binom{k}{j}-1}n^{j-k}$. Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov which makes use of a depth-first search to reveal the edges of a random graph. Our main original contribution is a "bounded degree lemma" which controls the structure of the component grown in the search process.
Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in which each of the ${n}\choose{d}$ possible edges is present with probability $p=p(n)$ independently, and let $H_d(n,m)$ denote a uniformly distributed with $n$ vertices and $m$ edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of $H_d(n,p)$ and $H_d(n,m)$ for the regime ${{n-1}\choose{d-1}} p,dm/n >(d-1)^{-1}+ε$. As an application, we obtain an asymptotic formula for the probability that $H_d(n,p)$ or $H_d(n,m)$ is connected. In addition, we infer a local limit theorem for the conditional distribution of the number of edges in $H_d(n,p)$ given connectivity. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.
Jaeger's directed cycle double cover conjecture can be formulated as a problem of existence of special perfect matchings in a class of graphs that we call hexagon graphs. In this work, we explore the structure of hexagon graphs. We show that hexagon graphs are braces that can be generated from the ladder on 8 vertices using two types of McCuaig's augmentations.
The Erdős-Rényi process begins with an empty graph on n vertices and edges are added randomly one at a time to a graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erdős and Rényi, various random graph models have been introduced and studied. In this paper we study the so-called Bohman-Frieze process, a simple modification of the Erdős-Rényi process. The Bohman-Frieze process begins with an empty graph on n vertices. At each step two random edges are present and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman-Frieze random graph process. We show that the Bohman-Frieze process has a qualitatively similar phase transition to the Erdős-Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c-\eps (that is, when the number of edges are (t_c-\eps) n/2) are trees or unicyclic components and that the largest component is of size Ω(\eps^{-2} \log n). Further, at t_c + \eps, all components apart from the giant component are trees or unicyclic and the size of the second-largest component is Θ(\eps^{-2} \log n). Each of these results corresponds to an analogous well-known result for the Erdős-Rényi process. Our methods include combinatorial arguments and a combination of the differential equation method for random processes with singularity analysis of generating functions which satisfy quasi-linear partial differential equations.
We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an "unbiased" way that a structure of size n gives rise to n pointed structures. We extend Polya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on Pólya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.
Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $π$ of the vertex set of $G$ into the plane. We prove that a wheel graph $W_n$ admits a drawing $π$ such that, if one wants to eliminate edge crossings by shifting vertices to new positions in the plane, then at most $(2+o(1))\sqrt n$ of all $n$ vertices can stay fixed. Moreover, such a drawing $π$ exists even if it is presupposed that the vertices occupy any prescribed set of points in the plane. Similar questions are discussed for other families of planar graphs.
Let $P(n,M)$ be a graph chosen uniformly at random from the family of all labeled planar graphs with $n$ vertices and $M$ edges. In the paper we study the component structure of $P(n,M)$. Combining counting arguments with analytic techniques, we show that there are two critical periods in the evolution of $P(n,M)$. The first one, of width $Θ(n^{2/3})$, is analogous to the phase transition observed in the standard random graph models and takes place for $M=n/2+O(n^{2/3})$, when the largest complex component is formed. Then, for $M=n+O(n^{3/5})$, when the complex components cover nearly all vertices, the second critical period of width $n^{3/5}$ occurs. Starting from that moment increasing of $M$ mostly affects the density of the complex components, not its size.
A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enumeration of the graphs embeddable on a given 2-dimensional surface (a main research topic of contemporary enumerative combinatorics) can also be formulated as the Gaussian matrix integral of an ice-type partition function. Some of the most puzzling conjectures of discrete mathematics are related to the notion of the cycle double cover. We express the number of the graphs with a fixed directed cycle double cover as the Gaussian matrix integral of an Ihara-Selberg-type function.
We prove that if one wants to make a plane graph drawing straight-line then in the worst case one has to move almost all vertices.
Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family of graphs (with some stability conditions) in terms of the 3-connected subfamily. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar. As a main application we recover in a purely combinatorial way the analytic expression found by Giménez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter.
We establish central and local limit theorems for the number of vertices in the largest component of a random $d$-uniform hypergraph $\hnp$ with edge probability $p=c/\binnd$, where $(d-1)^{-1}+\eps<c<\infty$. The proof relies on a new, purely probabilistic approach, and is based on Stein's method as well as exposing the edges of $H_d(n,p)$ in several rounds.
The logical depth of a graph $G$ is the minimum quantifier depth of a first order sentence defining $G$ up to isomorphism in the language of the adjacency and the equality relations. We consider the case that $G$ is a dissection of a convex polygon or, equivalently, a biconnected outerplanar graph. We bound the logical depth of a such $G$ from above by a function of combinatorial parameters of the dual tree of $G$.
We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number g_n of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and g_n is asymptotically $g n^{-5/2}ρ^{-n}$, where $g\approx0.00909941$ and $ρ^{-1}\approx7.50360$ can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on n vertices, for instance concerning connectedness, chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.