Related papers: An improved construction for the triangle removal …

In this paper we present improved results on the problem of counting triangles in edge streamed graphs. For graphs with $m$ edges and at least $T$ triangles, we show that an extra look over the stream yields a two-pass treaming algorithm…

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $\Delta(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph…

A well-known result of Nosal states that a graph $G$ with $m$ edges and $\lambda(G) > \sqrt{m}$ contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if $\lambda (G) >…

We investigate a covering problem in $3$-uniform hypergraphs ($3$-graphs): given a $3$-graph $F$, what is $c_1(n,F)$, the least integer $d$ such that if $G$ is an $n$-vertex $3$-graph with minimum vertex degree $\delta_1(G)>d$ then every…

A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e…

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$ such that…

We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class…

The Tur\'an problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $\Omega(n^{2-1/s})$ edges…

Let $G = (V, E)$ be a graph and $\lambda $ a non-negative integer. A graph $G$ is called a $(\lambda, 1)$-{\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\lambda$…

While the problem of determining whether an embedding of a graph $G$ in $\mathbb{R}^2$ is {\it infinitesimally rigid} is well understood, specifying whether a given embedding of $G$ is {\it rigid} or not is still a hard task that usually…

Chung and Graham [J. London Math. Soc. 1983] claimed to prove that there exists an $n$-vertex graph $G$ with $ \frac{5}{2}n \log_2 n + O(n)$ edges that contains every $n$-vertex tree as a subgraph. Frati, Hoffmann and T\'oth [Combin.…

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free…

A recent result of Eden, Levi, and Ron (ECCC 2015) provides a sublinear time algorithm to estimate the number of triangles in a graph. Given an undirected graph $G$, one can query the degree of a vertex, the existence of an edge between…

A well-known result of Mantel asserts that every $n$-vertex triangle-free graph $G$ has at most $\lfloor n^2/4 \rfloor$ edges. Moreover, Erd\H{o}s proved that if $G$ is further non-bipartite, then $e(G)\le \lfloor {(n-1)^2}/{4}\rfloor +1$.…

A graph is $c$-closed when every pair of nonadjacent vertices has at most $c-1$ common neighbors. In $c$-Closed Vertex Deletion, the input is a graph $G$ and an integer $k$ and we ask whether $G$ can be transformed into a $c$-closed graph…

In this paper, we introduce the generic circular triangle-free graph $\mathbb C_3$ and propose a finite axiomatization of its first order theory. In particular, our main results show that a countable graph $G$ embeds into $\mathbb C_3$ if…

A drawing of a graph is said to be a {\em straight-line drawing} if the vertices of $G$ are represented by distinct points in the plane and every edge is represented by a straight-line segment connecting the corresponding pair of vertices…

Graph Neural Networks (GNNs) as deep learning models working on graph-structure data have achieved advanced performance in many works. However, it has been proved repeatedly that, not all edges in a graph are necessary for the training of…

In 1979, Erd\H{o}s conjectured that if $m = O(n^{2/3})$, then $ex(n, m, \{C_4, C_6 \}) = O(n)$. This conjecture was disproven by several papers and the current best-known bounds for this problem are $$ c_1n^{1 + \frac{1}{15}} \leq ex(n,…

We study the following combinatorial counting and sampling problems: can we efficiently sample from the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ conditioned on triangle-freeness? Can we efficiently approximate the probability that $G(n,p)$…