Related papers: Tur\'{a}n problem for $\mathcal{K}_4^-$-free signe…

For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in…

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\rm{cd}(G)$ be the set of all character degrees of $G$ and denote by $\rho(G)$ the set of primes which divide some character…

The problem of determining the Tur\'an number of $C_4$ is a well studied problem that dates back to a paper of Erd\"os from 1938. It is known that Sidon sets can be used to construct $C_4$-free graphs. If $\A$ is a Sidon set in the abelian…

The spectral Tur\'an number $\spex(n, F)$ denotes the maximum spectral radius $\rho(G)$ of an $F$-free graph $G$ of order $n$. This paper determines $\spex\left(n, C_{2k+1}^{\square}\right)$ for all sufficiently large $n$, establishing the…

Given an $(r+1)$-chromatic graph $F$ and a graph $H$ that does not contain $F$ as a subgraph, we say that $H$ is strictly $F$-Tur\'an-good if the Tur\'an graph $T_{r}(n)$ is the unique graph containing the maximum number of copies of $H$…

We investigate natural Tur\'an problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural \textit{Tur\'an density coefficient} that measures how large a fraction of directed…

Let $\mathcal{F}$ be a set of graphs. The planar Tur\'an number, $ex_{\mathcal{P}}(n,\mathcal{F})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{F}$ as a subgraph. In this…

Given a family $\mathcal{F}$ of $r$-graphs, the Tur\'{a}n number of $\mathcal{F}$ for a given positive integer $N$, denoted by $ex(N,\mathcal{F})$, is the maximum number of edges of an $r$-graph on $N$ vertices that does not contain any…

Signed graphs are graphs whose edges get a sign $+1$ or $-1$ (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much…

The $kK_{r+1}$ is the union of $k$ disjoint copies of $(r+1)$-clique. Moon [Canad. J. Math. 20 (1968) 95--102] and Simonovits [Theory of Graphs (Proc. colloq., Tihany, 1996)] independently showed that if $n$ is sufficiently large, then…

This survey is two-fold. We first report new progress on the spectral extremal results on the Tur\'{a}n type problems in graph theory. More precisely, we shall summarize the spectral Tur\'{a}n function in terms of the adjacency spectral…

A graph $G$ is signed if each edge is assigned $+$ or $-$. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign $-$ if and only if its endpoints are in different parts. The Edwards-Erd\"os bound…

The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a subgraph. We…

A graph is called $K$-almost regular if its maximum degree is at most $K$ times the minimum degree. Erd\H{o}s and Simonovits showed that for a constant $0< \varepsilon< 1$ and a sufficiently large integer $n$, any $n$-vertex graph with more…

An oriented graph is a digraph obtained from an undirected graph by choosing an orientation for each edge. Given a positive integer $n$ and an oriented graph $F$, the oriented Tur$\acute{\rm a}$n number $ex_{ori}(n,F)$ is the maximum number…

For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence…

A graph $G$ is $k$-{\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. In a very recent paper, we gave a…

A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges…

Kostochka and Yancey resolved a famous conjecture of Ore on the asymptotic density of $k$-critical graphs by proving that every $k$-critical graph $G$ satisfies $|E(G)| \geq (\frac{k}{2} - \frac{1}{k-1})|V(G)| - \frac{k(k-3)}{2(k-1)}$. The…

This article foucuses on $(P_3\cup P_2,K_4)$-free graph. In this paper, we prove that if G is $(P_3\cup P_2,K_4)$-free, then $\chi(G)\le 7$. We then use our result to obtain the upper bound of order and chromatic number of…