Related papers: Rational values of the weak saturation limit

Let each vertex of a graph G = (V(G), E(G)) be given one of two colors, say, "black" and "white". Let Z denote the (initial) set of black vertices of G. The color-change rule converts the color of a vertex from white to black if the white…

Let $G$ be a graph with vertex set $V(G)$ and let $H:V(G)\rightarrow 2^N$ be a set function associating with $G$. An $H$-factor of graph $G$ is a spanning subgraphs $F$ such that $$d_F(v)\in H(v){4em}\hbox{for every}v\in V(G).$$ Let…

Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph…

Let $G$ be a simple $n$-vertex graph and $W\subseteq\V(G)$. We say that $W$ is a $\delta_k$-small set if $$ \sqrt[k]{\frac{\sum_{v\in W}d^k(v)}{\abs W}}\leq n-\abs W. $$ Let $\varphi^{(k)}(G)$ denote the smallest natural number $r$ such…

We study the parameterized problem of satisfying ``almost all'' constraints of a given formula $F$ over a fixed, finite Boolean constraint language $\Gamma$, with or without weights. More precisely, for each finite Boolean constraint…

In quantum theory, a weak value is a complex number with a somewhat technical definition: it is a ratio whose numerator is the matrix element of a self-adjoint operator and whose denominator is the inner product of a corresponding pair of…

For any simple graph $G$ on $n$ vertices, the (positive semi-definite) minimum rank of $G$ is defined to be the smallest possible rank among all (positive semi-definite) real symmetric $n\times n$ matrices whose entry in position $(i,j)$,…

We consider first-passage percolation on the class of "high-dimensional" graphs that can be written as an iterated Cartesian product $G\square G \square \dots \square G$ of some base graph $G$ as the number of factors tends to infinity. We…

Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to $e$ (including $e$ itself) is positive. Then the sum of weights over edges of $G$ is at least…

A graph is weakly $2$-colored if the nodes are labeled with colors black and white such that each black node is adjacent to at least one white node and vice versa. In this work we study the distributed computational complexity of weak…

The complexity of a reasoning task over a graphical model is tied to the induced width of the underlying graph. It is well-known that the conditioning (assigning values) on a subset of variables yields a subproblem of the reduced complexity…

We show that the following variation of the single-source shortest path problem is NP-complete. Let a weighted, directed, acyclic graph $G=(V,E,w)$ with source and sink vertices $s$ and $t$ be given. Let in addition a mapping $f$ on $E$ be…

We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into…

A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. An {\it edge-weighting} of $G$ is a function $w: E(G)\longrightarrow \mathbb{N}^+$, where $\mathbb{N}^+$ is the set of positive…

The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of…

A graph $G=(V,E)$ is representable if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. If $W$ is $k$-uniform (each letter of $W$ occurs exactly $k$…

Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly non-uniform) density of the point process.…

The weak degeneracy of a graph $G$ is a numerical parameter that was recently introduced by the first two authors with the aim of understanding the power of greedy algorithms for graph coloring. Every $d$-degenerate graph is weakly…

The weak $r$-coloring numbers $wcol_r(G)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $col(G)$, and have since found interesting theoretical and algorithmic applications. This has…

A vertex $v$ of a graph $G=(V,E)$ is said to be undefended with respect to a function $f: V \longrightarrow \{0,1,2\}$ if $f(v)=0$ and $f(u)=0$ for every vertex $u$ adjacent to $v$. We call the function $f$ a weak Roman dominating function…