It is shown that every Eulerian digraph on $n$ vertices with more than $(t-1)n$ arcs contains every oriented tree with $t$ edges. The digraphs have no loops or repeated arcs, but opposite arcs are permitted. The bound is sharp for each fixed oriented tree, as witnessed by disjoint unions of complete bidirected graphs. Previously, such tight bounds were not known, even just for directed paths. This can be considered as a directed analog of the recently proved Erd\H{o}s-S\'os conjecture. The result was proved by GPT-6 Astra.
A simplified proof of the Erd\H{o}s-S-S\'os conjecture on trees in graphs is presented, which is based on its recent resolution by GPT-6 Astra and characterized the extremal examples for Kalai's conjecture.
We show that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ vertex-disjoint even cycles with no edges between distinct cycles, or a set of $O(k^2\log k)$ vertices whose closed neighborhood meets every even cycle of $G$. Moreover, we give a polynomial-time algorithm that outputs one of the t...
Jungho Ahn, Lars Jaffke, O-joung Kwon et al.· 0 citations
Erd\H{o}s and S\'os conjectured that every $n$-vertex $3$-uniform hypergraph whose link graphs are all bipartite has at most $(1/4+o(1))\binom n3$ edges. We disprove this conjecture by constructing counterexamples with edge density at least $0.250000356>1/4$ for all sufficiently large $n$.
In the 1960s, Erd\H{o}s and Gallai conjectured that the edges of every $n$-vertex graph can be decomposed into $O(n)$ cycles and edges. We prove this conjecture. Equivalently, every Eulerian graph on $n$ vertices can be decomposed into $O(n)$ cycles, which confirms Haj\'os'conjecture up to a constant factor.
The Erd\H os--S\'os conjecture states that every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We prove that for every $\gamma$ there is an $n_0$ such that for all $n\ge n_0$ and $k \ge \gamma n$ the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old...
Erd\H{o}s, Frankl and R\"odl showed that the number of graphs on $n$ vertices with no copy of a non-bipartite graph $H$ is $2^{(1+o(1))ex(n,H)}$, whereas Morris and Saxton showed that this fails for $C_6$. Answering a question of Erd\H{o}s, we prove that the number of $C_4$-free graphs on $n$ vertices is $2^{(1+o(1))ex...
Tian-Chi Yang· 0 citations
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