For every $q\ge2$, we prove that every oriented graph $G$ on $n\ge45q-8$ vertices whose minimum semidegree satisfies \[ \delta^0(G)\ge \left\lceil\frac n3\right\rceil \] contains a directed cycle of length $3q$ through every vertex. The semidegree bound is sharp. This closes the one-unit gap left by the prescribed-vertex theorem of Kelly, K\"uhn and Osthus when $3\mid n$. We also prove that if an oriented graph $H$ has order $N$, minimum semidegree $d\ge3$, and $7d\ge2N+3$, then every ordered pair of distinct vertices is joined by a path of length three, four, or five. The constant $+3$ is best possible. As a consequence, the order hypothesis $n\ge10^{10}\ell$ in the general prescribed-vertex theorem of Kelly, K\"uhn and Osthus can be replaced by $n\ge15\ell-60$ for $\ell\ge7$.
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 a vertex $v$ of a graph $G$, the triangle-degree $\operatorname{td}_G(v)$ is the number of triangles containing $v$. A graph is triangle-distinct, or $K_3$-irregular, if its vertex triangle-degrees are pairwise distinct. Chartrand, Erd\H{o}s, and Oellermann asked whether a regular $K_3$-irregular graph exists. We p...
Let $\Ke$ be obtained from $K_7$ by deleting two independent edges. We prove that every 5-connected graph on $n\ge7$ vertices with at least $4n-9$ edges contains a $\Ke$ minor, settling Conjecture~1.4 of Dvo\v r\'ak, Norin and Rahman (arXiv preprint 2609.17760v1). The bound is sharp. We prove the stronger statement tha...
Caibing Chang, Zi-Jian Deng, Qin-Fei Tang et al.· 0 citations
For a graph $G$, let $f_t(G)$ denote the minimum of the maximum degree of an induced subgraph with $\alpha(G)+t$ vertices, where $\alpha(G)$ is the independence number, and write $f(G)=f_1(G)$. Huang's theorem gives $f(Q_k)\ge\lceil\sqrt{k}\rceil$ for the $k$-dimensional hypercube $Q_k$. We extend this lower bound to C...
Motivated by Mader's conjecture on connectivity keeping trees, we study trees obtained from paths by adding one pendant vertex, as well as related problems in triangle-free graphs. For an integer $m$ and $1\leq i\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex a...
Meng-Han Ma, Qing-Hai Liu, Li-Ping Zhang et al.· 0 citations
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 contai...
Geng-Tao Liu, Yun-Shu Gao· 0 citations
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