Related papers: Weak saturation stability

We show that for an infinitely many natural numbers $k$ there are $k$-uniform hypergraphs which admit a `rescaling phenomenon' as described in [9]. More precisely, let $\mathcal{A}(k,I, n)$ denote the class of $k$-graphs on $n$ vertices in…

A graph $G$ is called $F$-saturated if $G$ does not contain $F$ as a subgraph (not necessarily induced) but the addition of any missing edge to $G$ creates a copy of $F$. The saturation number of $F$, denoted by $sat(n,F)$, is the minimum…

In this paper, we study weakly dynamic undirected graphs, that can be used to represent some logistic networks. The goal is to deliver all the delivery points in the network. The network exists in a mostly stable environment, except for a…

We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. Using the small subgraph conditioning method, we prove an…

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a…

Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$ but the addition of any edge $e\notin E(G)$ creates at least one copy of…

We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(\omega)$ for some $n$. We…

The stability number of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size of G). A graph is alpha-stable if its stability number remains the same upon both the deletion and the addition of any…

A graph $G$ is weakly $\gamma$-closed if every induced subgraph of $G$ contains one vertex $v$ such that for each non-neighbor $u$ of $v$ it holds that $|N(u)\cap N(v)|<\gamma$. The weak closure $\gamma(G)$ of a graph, recently introduced…

In this work, we study the problem of stability of Graph Convolutional Neural Networks (GCNs) under random small perturbations in the underlying graph topology, i.e. under a limited number of insertions or deletions of edges. We derive a…

This paper delves into the stability of the $2$-domination number in simple undirected graphs. The $2$-domination number of a graph $G$, $\gamma_2(G)$, represents the minimum size of a vertex subset where every other vertex in the graph is…

Fix graphs $F$ and $H$. Let $\mathrm{ex}(n,H,F)$ denote the maximum number of copies of a graph $H$ in an $n$-vertex $F$-free graph. In this note we will give a new general supersaturation result for $\mathrm{ex}(n,H,F)$ in the case when…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $d n$ is bounded by a constant depending only on $H$ and $d$.…

For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum…

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary fixed graph $H$. We prove that, for all but finitely many values of the edge density, if the…

Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…

For integers $k>\ell\ge0$, a graph $G$ is $(k,\ell)$-stable if $\alpha(G-S)\geq \alpha(G)-\ell$ for every $S\subseteq V(G)$ with $|S|=k$. A recent result of Dong and Wu [SIAM J. Discrete Math., 36 (2022) 229--240] shows that every…

In this paper, we study the complete graph $K_n$ with n vertices, where we attach an independent and identically distributed (i.i.d.) weight to each of the n(n-1)/2 edges. We focus on the weight $W_n$ and the number of edges $H_n$ of the…

Stability and dependence are model-theoretic notions that have recently proved highly effective in the study of structural and algorithmic properties of hereditary graph classes, and are considered key notions for generalizing to hereditary…