Related papers: Universality for graphs with bounded density

We consider the next greedy randomized process for generating maximal H-free graphs: Given a fixed graph H and an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph. Then, construct an…

Let $G$ be a finite insoluble group with soluble radical $R(G)$. In this paper we investigate the soluble graph of $G$, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in $G…

Let G be a weighted (directed) graph with n vertices and m edges. Given a source vertex s, Dijkstra's algorithm computes the shortest path lengths from s to all other vertices in O(m + n log n) time. This bound is known to be worst-case…

We show that there exists a graph $G$ with $O(n)$ nodes, where any forest of $n$ nodes is a node-induced subgraph of $G$. Furthermore, for constant arboricity $k$, the result implies the existence of a graph with $O(n^k)$ nodes that…

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complex $\Delta$ is called pure if all of its facets have the same cardinality. Let $\mathcal G$ be the class of graphs…

For an $n$-vertex graph $G$, let $h(G)$ denote the smallest size of a subset of $V(G)$ such that it intersects every maximum independent set of $G$. A conjecture posed by Bollob\'{a}s, Erd\H{o}s and Tuza in early 90s remains widely open,…

Inspired by a famous characterization of perfect graphs due to Lov\'{a}sz, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $\alpha(H) + \omega(H) \geq |V(H)|$. (Here $\alpha$ and $\omega$ denote the…

The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that,…

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H…

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…

P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We…

Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint…

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar if every edge is crossed at most k times and RAC if edges can…

Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e…

Denote by $r_g(G,\mathcal{H})$ the global resilience of a graph $G$ with respect to Hamiltonicity. That is, $r_g(G,\mathcal{H})$ is the minimal $r$ for which there exists a subgraph $H\subseteq G$ with $r$ edges, such that $G\setminus H$ is…

Due to the increasing discovery and implementation of networks within all disciplines of life, the study of subgraph connectivity has become increasingly important. Motivated by the idea of community (or sub-graph) detection within a…

Let $\mathcal{H}$ be a $k$-uniform hypergraph. A chain in $\mathcal{H}$ is a sequence of its vertices such that every $k$ consecutive vertices form an edge. In 1999 Katona and Kierstead suggested to use chains in hypergraphs as the…

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for…