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Let $G$ be a graph (with multiple edges allowed) and let $T$ be a tree in $G$. We say that $T$ is $\textit{even}$ if every leaf of $T$ belongs to the same part of the bipartition of $T$, and that $T$ is $\textit{weakly even}$ if every leaf of $T$ tha
Externí odkaz:
http://arxiv.org/abs/2409.15522
Autor:
Pokrovskiy, Alexey
A rough structure theorem is proved for graphs $G$ containing no copy of a bounded degree tree $T$: from any such $G$, one can delete $o(|G||T|)$ edges in order to get a subgraph all of whose connected components have a cover of order $3|T|$. This th
Externí odkaz:
http://arxiv.org/abs/2409.15191
Autor:
Pokrovskiy, Alexey
This is a companion paper to the paper "Hyperstability in the Erdos-Sos Conjecture". In that paper the following rough structure theorem was proved for graphs G containing no copy of a bounded degree tree T: from any such G, one can delete o(|G||T|)
Externí odkaz:
http://arxiv.org/abs/2409.15189
Autor:
Finch, Steven
We treat three cubic recurrences, two of which generalize the famous iterated map $x \mapsto x (1-x)$ from discrete chaos theory. A feature of each asymptotic series developed here is a constant, dependent on the initial condition but otherwise intri
Externí odkaz:
http://arxiv.org/abs/2409.15175
Given a graph $G$ and a family of graphs $\cal F$, an $\cal F$-isolating set, as introduced by Caro and Hansberg, is any set $S\subset V(G)$ such that $G - N[S]$ contains no member of $\cal F$ as a subgraph. In this paper, we introduce a game in whic
Externí odkaz:
http://arxiv.org/abs/2409.14180
We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a node as the
Externí odkaz:
http://arxiv.org/abs/2409.13472
Autor:
Tom, Foster
We prove that a tree with a vertex of degree at least five must be missing a connected partition of some type and therefore its chromatic symmetric function cannot be $e$-positive. We prove that this also holds for a tree with a vertex of degree four
Externí odkaz:
http://arxiv.org/abs/2409.12934
Given a finite set of red and blue points in $\mathbb{R}^d$, the MST-ratio is the combined length of the Euclidean minimum spanning trees of red points and of blue points divided by the length of the Euclidean minimum spanning tree of the union of th
Externí odkaz:
http://arxiv.org/abs/2409.11079
Autor:
Lyons, Benjamin, Olsen, McCabe
We study lattice polytopes which arise as the convex hull of chip vectors for \textit{self-reachable} chip configurations on a tree $T$. We show that these polytopes always have the integer decomposition property and characterize the vertex sets of t
Externí odkaz:
http://arxiv.org/abs/2409.07675
Autor:
Carter, Daniel
A family of graphs $\mathcal{F}$ is said to have the joint embedding property (JEP) if for every $G_1, G_2\in \mathcal{F}$, there is an $H\in \mathcal{F}$ that contains both $G_1$ and $G_2$ as induced subgraphs. If $\mathcal{F}$ is given by a finite
Externí odkaz:
http://arxiv.org/abs/2409.06127