Related papers: Classification of OBDD size for monotone 2-CNFs

A {\it 2-rainbow domination function} of a graph $G$ is a function $f$ that assigns to each vertex a set of colors chosen from the set $\{1,2\}$, such that for any $v\in V(G)$, $f(v)=\emptyset$ implies $\bigcup_{u\in N(v)}f(u)=\{1,2\}$. The…

For a graph $G$, let $a(G)$ denote the maximum size of a subset of vertices that induces a forest. We prove the following. 1. Let $G$ be a graph of order $n$, maximum degree $\Delta>0$ and maximum clique size $\omega$. Then \[ a(G) \geq…

In the last decade, algorithmic frameworks based on a structural graph parameter called mim-width have been developed to solve generally NP-hard problems. However, it is known that the frameworks cannot be applied to the Clique problem, and…

The scramble number of a graph provides a lower bound for gonality and an upper bound for treewidth, making it a graph invariant of interest. In this paper we study graphs of scramble number at most two, and give a classification of all…

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomass\'e…

Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show…

We use Razborov's flag algebra method to show an asymptotic upper bound for the maximal induced density $i(\vec P_3)$ of the orgraph $\vec P_3$ in an arbitrary orgraph. A conjecture of Thomass\'e states that $i(\vec P_3)=2/5$. The hitherto…

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface $\Sigma$ and integers $\Delta$ and $k$, determine the maximum order $N(\Delta,k,\Sigma)$ of a graph embeddable in $\Sigma$ with maximum degree…

For a finite graph $G$, we study the maximum $2$-edge colorable subgraph problem and a related ratio $\frac{\mu(G)}{\nu(G)}$, where $\nu(G)$ is the matching number of $G$, and $\mu(G)$ is the size of the largest matching in any pair…

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with the same color. The graph $G$ is called monochromatic disconnected if any two distinct vertices of $G$ are separated by a…

Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$ and vertex conductance $\Psi^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap…

A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident…

We present a data stream algorithm for estimating the size of the maximum matching of a low arboricity graph. Recall that a graph has arboricity $\alpha$ if its edges can be partitioned into at most $\alpha$ forests and that a planar graph…

The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether…

We provide a characterization of graphs of linear rankwidth at most 1 by minimal excluded vertex-minors.

Given a graph $G$, the maximum size of an induced subgraph of $G$ each component of which is a star is called the edge open packing number, $\rho_{e}^{o}(G)$, of $G$. Similarly, the maximum size of an induced subgraph of $G$ each component…

We investigate the ratio $\avM(G)$ of the average size of a maximal matching to the size of a maximum matching in a graph $G$. If many maximal matchings have a size close to $\maxM(G)$, this graph invariant has a value close to 1.…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

We study the number of lines in hypergraphs in a more symmetric setting, where both the hypergraph and its complement are considered. In the general case and in some special cases, the lower bounds on the number of lines are much higher…

The orthogonality dimension of a graph $G$ over $\mathbb{R}$ is the smallest integer $k$ for which one can assign a nonzero $k$-dimensional real vector to each vertex of $G$, such that every two adjacent vertices receive orthogonal vectors.…