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Let $A in mathbb{R}^{n times n}$ be the adjacency matrix of an ErdH{o}s Renyi graph $G(n, d/n)$ for $d = omega(1)$ and $d leq 3log(n)$. We show that as $n$ goes to infinity, with probability that goes to $1$, the adjacency matrix of the $3$-core of $G(n, d/n)$ is invertible.
Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of ErdH{o}s-Renyi random graphs $G(n, p_n)$, where $p_n = n^{-alpha}$ for $0 < alpha < 1$. We
Given an unlabeled graph $G$ on $n$ vertices, let ${N_{G}(v)}_{v}$ be the collection of subgraphs of $G$, where for each vertex $v$ of $G$, $N_{G}(v)$ is the subgraph of $G$ induced by vertices of $G$ of distance at most one from $v$. We show that th
A (not necessarily proper) vertex colouring of a graph has clustering $c$ if every monochromatic component has at most $c$ vertices. We prove that planar graphs with maximum degree $Delta$ are 3-colourable with clustering $O(Delta^2)$. The previous b
Let $G$ be an $n$-vertex graph and let $L:V(G)rightarrow P({1,2,3})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can determine $L$-
We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of