A digraph is antidirected if every vertex has indegree zero or outdegree zero. Let $k\ge2$, and let $F$ be an antidirected forest with $k$ arcs and no isolated vertices. We prove that every digraph $D$ of order $n$ with more than $g_k(n):=2\max\left\{\binom{2k-1}{2}, (k-1)\left(n-\frac{k}{2}\right)\right\}$ arcs contains $F$ as a subdigraph. For $n\ge2k-1$, this threshold equals $2\mathrm{ex}(n,kK_2)$ and is attained by symmetric digraphs arising from extremal $kK_2$-free graphs. Consequently, the maximum directed extremal number over all such forests is $2\mathrm{ex}(n,kK_2)$. The proof combines a counting inequality for rooted antidirected forests, embeddings extending vertex-disjoint arcs, and vertex deletion. In the remaining case, the Gallai--Edmonds decomposition of the underlying graph gives the required bound on the number of arcs.
Let $G$ be a graph together with a total order $\prec$ on its edges. We say that $\prec$ is realizable in $\mathbb{R}^d$ if there is a placement of the vertices of $G$ in $\mathbb{R}^d$ such that the Euclidean lengths of the edges induce exactly the order $\prec$. Almendra-Hern\'andez and Mart\'inez-Sandoval proved tha...
Gerardo L. Maldonado, Leonardo Martínez-Sandoval, Miguel Raggi et al.· 0 citations
For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)...
Fix an integer $d\ge1$. We prove that, for all sufficiently large $m$, every digraph $D$ with $m$ arcs and minimum semidegree at least $d$ admits a bisection $V(D)=V_1\cup V_2$ with $\bigl||V_1|-|V_2|\bigr|\le1$ such that \[ \min\{e(V_1,V_2),e(V_2,V_1)\} \ge \frac{d(m+d+1)}{2(2d+1)}. \] Here, for disjoint vertex sets $...
Let $\mathbf{G}=\{G_1,\dots,G_{n}\}$ be a collection of not necessarily distinct $n$-vertex graphs with a common vertex set $V$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq n$ is called \emph{rainbow} in $\mathbf{G}$, if there exists an injection $\phi\colon E(C)\to [n]$ such that $e\in E(G_{\phi(e)})$ for each...
For a graph $H$, the anti-Ramsey number $\operatorname{ar}(n,H)$ is the maximum number of colors in an edge-coloring of $K_n$ containing no rainbow copy of $H$, where a copy is rainbow if its edges have pairwise distinct colors. Let $s,t$ be nonnegative integers with $s+t\ge2$, and let $H_{s,t}$ be a graph consisting o...
For a fixed digraph $F$, let $\operatorname{ex}_2^+(n,F)$ be the maximum of $\sum_{v\in V(D)}d_D^+(v)^2$ over all $n$-vertex $F$-free digraphs. Ai et al. [arXiv:2606.03520, 2026] asked for which self-converse tournament $F$ one can determine $\operatorname{ex}_2^+(n,F)$. Let $TT_r$ denote a transitive tournament on $r$...
Zhuo-Ran Han, Yao-Jun Chen· 0 citations
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