Journal of Combinatorial Theory Series B最新文献

Let T be an oriented tree on n vertices with maximum degree at most $e^{o\left(\sqrt{\log n}\right)}$. If G is a digraph on n vertices with minimum semidegree ${\delta }^0\left(G\right)\ge (\frac{1}{2}+o(1\left)\right)n$, then G contains T as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree $o(n/\log n)$). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of T the digraph G contains. Our main result states that every such G contains at least $|\mathrm{Aut}(T){|}^{-1}{(\frac{1}{2}-o(1\left)\right)}^{n}n!$ copies of T, which is optimal. This implies the analogous result in the undirected case.

The burning number $b\left(G\right)$ of a graph G is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that $b\left(G\right)\le \lceil \sqrt{n}\rceil$ for all connected graphs G on n vertices. We prove that this conjecture holds asymptotically, that is $b\left(G\right)\le (1+o(1\left)\right)\sqrt{n}$.

For a finite abelian group A, define $f\left(A\right)$ to be the minimum integer such that for every complete digraph Γ on f vertices and every map $w:E\left(\Gamma \right)\to A$, there exists a directed cycle C in Γ such that ${\sum }_{e\in E\left(C\right)}w\left(e\right)=0$. The study of $f\left(A\right)$ was initiated by Alon and Krivelevich (2021). In this article, we prove that $f\left(Z_p^k\right)=O\left(pk{(\log k)}^2\right)$, where p is prime, with an improved bound of $O(k\log k)$ when $p=2$. These bounds are tight up to a factor which is polylogarithmic in k.

The famous Dirac's Theorem gives an exact bound on the minimum degree of an n-vertex graph guaranteeing the existence of a hamiltonian cycle. In the same paper, Dirac also observed that a graph with minimum degree at least $k\ge 2$ contains a cycle of length at least $k+1$. The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least k in r-uniform, n-vertex hypergraphs for all combinations of $k,r$ and n with $3\le r,k\le n$. The bounds differ for different ranges of r compared to n and k.

A graph is ℓ-holed if all its induced cycles of length at least four have length exactly ℓ. We give a complete description of the ℓ-holed graphs for each $\ell \ge 7$.

We prove that the only trees that admit perfect state transfer according to the adjacency matrix model are $P_2$ and $P_3$. This answers a question first asked by Godsil in 2012 and proves a conjecture by Coutinho and Liu from 2015.

Scott and Seymour conjectured the existence of a function $f:N\to N$ such that, for every graph G and tournament T on the same vertex set, $\chi \left(G\right)⩾f\left(k\right)$ implies that $\chi \left(G\right[N_T^{+}\left(v\right)\left]\right)⩾k$ for some vertex v. In this note we disprove this conjecture even if v is replaced by a vertex set of size $O(\log |V\left(G\right)\left|\right)$. As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomassé, and Wu concerning the corresponding statement where the graph G is replaced by another tournament, and disprove a related conjecture of Nguyen, Scott, and Seymour. We also show that the setting where chromatic number is replaced by degeneracy exhibits a quite different behaviour.

In 1995, Komlós, Sárközy and Szemerédi showed that every large n-vertex graph with minimum degree at least $(1/2+\gamma )n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all γ and Δ, and n large, every n-vertex 3-uniform hypergraph of minimum vertex degree $(5/9+\gamma )\left(\begin{array}{c}n\\ 2\end{array}\right)$ contains every loose spanning tree T with maximum vertex degree Δ. This bound is asymptotically tight, since some loose trees contain perfect matchings.

We consider the problem of maximizing the number of induced copies of an oriented star $S_{k,\ell }$ in digraphs of given size, where the center of the star has out-degree k and in-degree ℓ. The case $k\ell =0$ was solved by Huang in [11]. Here, we asymptotically solve it for all other oriented stars with at least seven vertices.

Two subsets in a group are called twins if each is contained in a left translate of the other, though the two sets themselves are not translates of each other. We show that in the free group $F_{\{\alpha ,\beta \}}$, there exist maximal families of twins of any finite cardinality. This result is used to show that in the context of embeddings of trees, there exist maximal families of twin trees of any finite cardinality. These are counterexamples to the “tree alternative” conjecture, which supplement the first counterexamples published by Kalow, Laflamme, Tateno, and Woodrow. We also investigate twin sets in the sphere $S_2$, where the embeddings considered are isometries of $S_2$. We show that there exist maximal families of twin sets in $S_2$ of any finite cardinality.