Related papers: Even-degeneracy of a random graph

We study the algorithmic decidability of the domination number in the Erdos-Renyi random graph model $G(n,p)$. We show that for a carefully chosen edge probability $p=p(n)$, the domination problem exhibits a strong irreducible property.…

Conventionally used exponential random graphs cannot directly model weighted networks as the underlying probability space consists of simple graphs only. Since many substantively important networks are weighted, this limitation is…

We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al.: If $G$ is a graph of maximum degree…

We study a random even subgraph of a finite graph $G$ with a general edge-weight $p\in(0,1)$. We demonstrate how it may be obtained from a certain random-cluster measure on $G$, and we propose a sampling algorithm based on coupling from the…

We show that almost surely the rank of the adjacency matrix of the Erd\"os-R\'enyi random graph $G(n,p)$ equals the number of non-isolated vertices for any $c\ln n/n<p<1/2$, where $c$ is an arbitrary positive constant larger than 1/2. In…

The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \epsilon>0 and p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while…

Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be…

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well.…

We study the maximum dimension $d=d(n,p)$ for which an Erd\H{o}s-R\'enyi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point…

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a…

For a graph $G=(V,E)$, let $\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\tau(G) \leq…

We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex…

A graph is \emph{$(\mathcal{I}, \mathcal{F})$-partitionable} if its vertex set can be partitioned into two parts such that one part $\mathcal{I}$ is an independent set, and the other $\mathcal{F}$ induces a forest. A graph is…

We prove the following 30-year old conjecture of Gy\H{o}ri and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\ldots,C_\ell$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le…

Bernshteyn and Lee defined a new notion, weak degeneracy, which is slightly weaker than the ordinary degeneracy. It is proved that strictly $f$-degenerate transversal is a common generalization of list coloring, $L$-forested-coloring and…

Given $D$ and $\gamma>0$, whenever $c>0$ is sufficiently small and $n$ sufficiently large, if $\mathcal{G}$ is a family of $D$-degenerate graphs of individual orders at most $n$, maximum degrees at most $\tfrac{cn}{\log n}$, and total…

We deal with a random graph model where at each step, a vertex is chosen uniformly at random, and it is either duplicated or its edges are deleted. Duplication has a given probability. We analyse the limit distribution of the degree of a…

A hypergraph is "$d$-degenerate" if every subhypergraph has a vertex of degree at most $d$. A greedy algorithm colours every such hypergraph with at most $d+1$ colours. We show that this bound is tight, by constructing an $r$-uniform…

In this paper, we consider the Tur\'an problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and…

The exponential family of random graphs has been a topic of continued research interest. Despite the relative simplicity, these models capture a variety of interesting features displayed by large-scale networks and allow us to better…