Related papers: Graphs with induced-saturation number zero

A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of…

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$,…

The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical…

As introduced by Bollob\'as, a graph $G$ is weakly $H$-saturated if the complete graph $K_n$ is obtained by iteratively completing copies of $H$ minus an edge. For all graphs $H$, we obtain an asymptotic lower bound for the critical…

In this paper we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph $H$ and an integer $t$, what is $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$, the minimum number of edges in a…

Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from…

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f: E(H)\rightarrow…

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by…

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$).…

A graph $G$ is uniquely $H$-saturated if it contains no copy of a graph $H$ as a subgraph, but adding any new edge into $G$ creates exactly one copy of $H$. Let $C_{4}^{+}$ be the diamond graph consisting of a $4$-cycle $C_{4}$ with one…

We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for…

In 1984, Erd\H{o}s and Simonovits conjectured the following: given a bipartite graph $H$, there exist constants $\beta, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C \mathrm{ex}(n, H)$ edges contains at least $\beta…

We say that an edge-coloring of a graph $G$ is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of $G$ receive the same color. Furthermore, given a fixed graph $F$, we say that $G$ is rainbow…

Given a $k$-graph $H$ a complete blow-up of $H$ is a $k$-graph $\hat{H}$ formed by replacing each $v\in V(H)$ by a non-empty vertex class $A_v$ and then inserting all edges between any $k$ vertex classes corresponding to an edge of $H$.…

We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.

An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and…

Given a graph $H$, we say a graph $G$ is properly rainbow $H$-saturated if there is a proper edge-coloring of $G$ which contains no rainbow copy of $H$, but adding any edge to $G$ makes such an edge-coloring impossible. The proper rainbow…

An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and…

Subdividing an edge $uv$ in a graph replaces it by a path $u w v$ with one new vertex. For a graph $H$, the \textsc{$H$-free Subdivision} problem asks whether, given a graph $G$ and an integer $k$, one can destroy all induced copies of $H$…

An induced matching $M$ in a graph $G$ is dominating if every edge not in $M$ shares exactly one vertex with an edge in $M$. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph $G$ contains…