Related papers: Results on Colored Tree Properties
We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree $T$ of order $n$, $cfc(T)\geq cfc(P_n)=\lceil \log_{2}…
Teminal Wiener index is one of the commonly used topological index in mathematical chemistry. If two or more chemical compounds have the same terminal Wiener index then they will have similar physico-chemical properties. In this work we…
The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for…
We provide a relatively compact proof of the BPHZ theorem for regularity structures of decorated trees in the case where the driving noise satisfies a suitable spectral gap property, as in the Gaussian case. This is inspired by the recent…
This paper is a continuation of arXiv:1612.03873. We prove a three-parameter family of identities (Theorem 1.1) involving a version of the Tutte polynomial for directed graphs introduced by Awan and Bernardi in arXiv:1610.01839. A…
The paper deals with two issues: the existence of universal models of a theory T and related properties when cardinal arithmetic does not give this existence offhand. In the first section we prove that simple theories (e.g., theories…
The entropy of a graph is an information-theoretic quantity which expresses the complexity of a graph \cite{DM1,M}. After Shannon introduced the definition of entropy to information and communication, many generalizations of the entropy…
Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey…
Measures of tree balance play an important role in various research areas, for example in phylogenetics. There they are for instance used to test whether an observed phylogenetic tree differs significantly from a tree generated by the Yule…
We consider a Ramsey statement for pairs of maps between trees, where one is an embedding as defined by Deuber and the other is a rigid surjection as defined by Solecki. We show that there is no Ramsey Theorem for pairs of maps where the…
We consider the restriction of Ramsey's theorem that arises from considering only translation-invariant colourings of pairs, and show that this has the same strength (both from the viewpoint of Reverse Mathematics and from the viewpoint of…
The concept of trustworthy AI has gained widespread attention lately. One of the aspects relevant to trustworthy AI is robustness of ML models. In this study, we show how to probabilistically quantify robustness against naturally occurring…
We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Ne\v{s}et\v{r}il-R\"{o}dl Theorem, the author's Ramsey lift of bowtie-free graphs as well…
We study a wide class of topological free-fermion systems on a hypercubic lattice in spatial dimensions $d\ge 1$. When the Fermi level lies in a spectral gap or a mobility gap, the topological properties, e.g., the integral quantization of…
Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an…
To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the…
We present the Chromatic Persistence Algorithm (CPA), an event-driven method for computing persistent cohomological features of weighted graphs via graphic arrangements, a classical object in computational geometry. We establish rigorous…
Structures of multilinear maps are characterized by invariants. In this paper we introduce two invariants, named the isotropy index and the completeness index. These invariants capture the tensorial structure of the kernel of a multilinear…
Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k = 0}^n (-1)^k c_k \lambda^{n - k}$. Laplacian--like energy of a graph is newly proposed…
This work continues to investigate the link between differentially private (DP) and online learning. Alon, Livni, Malliaris, and Moran (2019) showed that for binary concept classes, DP learnability of a given class implies that it has a…