Related papers: Spectral supersaturation: Triangles and bowties

The $3$-path isolation number of a connected $n$-vertex graph $G$, denoted by $\iota(G,P_3)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$ intersects the vertex sets…

A vertex set $S$ is a generalized $k$-independent set if the induced subgraph $G[S]$ contains no tree on $k$ vertices. The generalized $k$-independence number $\alpha_k(G)$ is the maximum size of such a set. For a tree $T$ with $n$…

An edge-coloring of a graph $H$ is a function $\mathcal{C}: E(H) \rightarrow \mathbb{N}$. We say that $H$ is rainbow if all edges of $H$ have different colors. Given a graph $F$, an edge-colored graph $G$ is $F$-rainbow saturated if $G$…

Let $G$ be a graph and $\mathcal{F}$ be a family of graphs. We say a graph $G$ is $\mathcal{F}$-saturated if $G$ does not contain any member in $\mathcal{F}$ and for any $e\in E(\overline{G})$, $G+e$ creates a copy of some member in $…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.

For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all…

An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhood and $S$. It was conjectured that there exists a…

Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $\delta^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot…

In 1975, Erd\H{o}s and Sauer asked to estimate, for any constant $r$, the maximum number of edges an $n$-vertex graph can have without containing an $r$-regular subgraph. In a recent breakthrough, Janzer and Sudakov proved that any…

Let $G$ be a $K_4$-free graph, an edge in its complement is a $K_4$-\emph{saturating} edge if the addition of this edge to $G$ creates a copy of $K_4$. Erd\H{o}s and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with…

An edge coloring of a graph $G$ is a Gallai coloring if it contains no rainbow triangle. We show that the number of Gallai $r$-colorings of $K_n$ is $\left(\binom{r}{2}+o(1)\right)2^{\binom{n}{2}}$. This result indicates that almost all…

Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some…

Given an integer $r\ge1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A…

Extremal problems on the $4$-cycle $C_4$ played a heuristic important role in the development of extremal graph theory. A fundamental theorem of F\"uredi states that the Tur\'an number $ex(q^2+q+1, C_4)\leq \frac12 q(q+1)^2$ holds for every…

A well-known theorem of Erd\H{o}s and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Gy\H{o}ri, Katona and Lemons gave an extension of this result to hypergraphs by determining…

In 1989, Zehavi and Itai conjectured that every $k$-connected graph contains $k$ independent spanning trees rooted at any prescribed vertex $r$. That is, for each vertex $v$, the unique $r$-$v$ paths within these $k$ spanning trees are…

A chord of a cycle $C$ is an edge joining two non-consecutive vertices of $C$. A cycle $C$ in a graph $G$ is chorded if the vertex set of $C$ induces at least one chord. In this paper, we prove that if $G$ is a graph with order $n\geq 6$…

It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we…

A dominating set D of a graph G is a set such that each vertex v of G is either in the set or adjacent to a vertex in the set. Matheson and Tarjan (1996) proved that any n-vertex plane triangulation has a dominating set of size at most n/3,…