Related papers: Non-bipartite graphs without theta subgraphs

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…

Given a family of graphs $\mathcal{H}$, the extremal number $\textrm{ex}(n, \mathcal{H})$ is the largest $m$ for which there exists a graph with $n$ vertices and $m$ edges containing no graph from the family $\mathcal{H}$ as a subgraph. We…

A graph $G$ is called $H$-free, if it does not contain $H$ as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Tur\'{a}n type problem: what is the maximum spectral radius of an $H$-free graph of order $n$? In this paper, we…

Suppose that $R$ (red) and $B$ (blue) are two graphs on the same vertex set of size $n$, and $H$ is some graph with a red-blue coloring of its edges. How large can $R$ and $B$ be if $R\cup B$ does not contain a copy of $H$? Call the largest…

Given graphs $H_1, H_2$, a {red, blue}-coloring of the edges of a graph $G$ is a critical coloring if $G$ has neither a red $H_1$ nor a blue $ H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G$ admits a critical coloring, but…

Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Tur\'an number} $ex(n,\mc{F})$, the maximum…

Let the bipartite Tur\'an number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any…

A stability result due to Ren, Wang, Wang and Yang [SIAM J. Discrete Math. 38 (2024)] shows that if $3\le r \le 2k$ and $n\ge 318 (r-2)^2k$, and $G$ is a $C_{2k+1}$-free graph on $n$ vertices with $e(G)\ge \lfloor {(n-r+1)^2}/{4}\rfloor +{r…

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$ denote the maximum value of $\mathcal{P}$ over all $n$-vertex graphs with…

Erd\H{o}s and Simonovits asked the following question: For an integer $r\geq 2$ and a family of non-bipartite graphs $\mathcal{H}$, determine the infimum of $\alpha$ such that any $\mathcal{H}$-free $n$-vertex graph with minimum degree at…

Nikiforov conjectured that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ (or $\mu(G)\ge \mu(S_{n,k}^+))$ contains $C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless…

Given a graph family $\mathbb{H}$, let ${\rm SPEX}(n,\mathbb{H}_{\rm sub})$ denote the set of $n$-vertex $\mathbb{H}$-subdivision-free graphs with the maximum spectral radius. In this paper, we investigate the problem of graph subdivision…

The \emph{Tur\'an function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key…

The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines the exact bound on the number of edges $t_r(n)$ in an $n$-vertex graph that does not contain a clique of size $r+1$. We establish an interesting link between…

Let $F$ be a graph with chromatic number $\chi(F) = r+1$. Denote by $ex(n, F)$ and $Ex(n, F)$ the Tur\'{a}n number and the set of all extremal graphs for $F$, respectively. In addition, $ex_{ssp}(n, F)$ and $Ex_{ssp}(n, F)$ are the maximum…

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erd\H{o}s and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close…

A well-known theorem of Vizing states that if $G$ is a simple graph with maximum degree $\Delta$, then the chromatic index $\chi'(G)$ of $G$ is $\Delta$ or $\Delta+1$. A graph $G$ is class 1 if $\chi'(G)=\Delta$, and class 2 if…

The classical spectral Tur\'{a}n problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n$. Zhai and Wang [Linear Algebra Appl, 437 (2012) 1641-1647] determined the maximum spectral radius of…

The generalized Tur\'an number $\text{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $kK_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner…

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a…