Related papers: An improved upper bound on the oriented diameter o…
We consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed, undirected graph. We are motivated by the task of deriving lower bounds for consensus protocols and by the so-called "definitive…
Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs by Chv\'{a}tal and Hanson (1976), and by Balachandran and Khare (2009). It follows from the structure of those extremal…
Let $G$ be a finite, connected graph. The eccentricity of a vertex $v$ of $G$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the arithmetic mean of the eccentricities of the vertices of $G$. We…
We consider the following long range percolation model: an undirected graph with the node set $\{0,1,...,N\}^d$, has edges $(\x,\y)$ selected with probability $\approx \beta/||\x-\y||^s$ if $||\x-\y||>1$, and with probability 1 if…
A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph $G$, let $\beta(G)$ denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion…
For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function $f$ such that if the tree-width of a graph $G$ with $k$ pairs of terminals is at least $f(k)$, then $G$ contains a…
Given any two vertices u, v of a random geometric graph, denote by d_E(u,v) their Euclidean distance and by d_G(u,v) their graph distance. The problem of finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of attention…
In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic functions with leading coefficient 1/4…
In 1941, Turan conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many…
In this paper, we study the oriented diameter of power graphs of groups. We show that a $2$-edge connected power graph of a finite group has oriented diameter at most $4$. We prove that the power graph of the cyclic group of order $n$ has…
In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the…
We give an improved lower bound of 10/3 on the competitive ratio for the exploration of an undirected, edge-weighted graph with a single agent that needs to return to the starting location after visiting all vertices. We assume that the…
The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances).…
For an oriented graph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$. Let $f(n)$ be the smallest integer such that every oriented graph $G$ with chromatic number larger than $f(n)$ has $f(G) > n$. Let…
Let $s(n)$ be the minimum number of edges in a graph that contains every $n$-vertex tree as a subgraph. Chung and Graham [J. London Math. Soc. 1983] claim to prove that $s(n)\leqslant O(n\log n)$. We point out a mistake in their proof. The…
A graph $U$ is an induced universal graph for a family $F$ of graphs if every graph in $F$ is a vertex-induced subgraph of $U$. For the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an…
An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The…
An "edge guard set" of a plane graph $G$ is a subset $\Gamma$ of edges of $G$ such that each face of $G$ is incident to an endpoint of an edge in $\Gamma$. Such a set is said to guard $G$. We improve the known upper bounds on the number of…
A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the…
In 1975, P. Erd\"os proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq…