arXiv subjects
Publications and source records attributed to Pavel Prozorov.
Suppose that $n=2m$, $k=2t$ and $n > 10 k^7$. We show that if a family $\mathcal F$ of $k$-subsets of an $n$-set has only even pairwise intersections then $|\mathcal F| \leq \binom{m}{t}$. Moreover, every extremal family has an atomic structure. This result was previously proved by Frankl and Tokushige for $n > n_{FT}(k)$, where $n_{FT}(k)$ is at least exponential. The main technique is Delsarte linear programming.
The celebrated Steiner tree problem is the problem of finding a set $St$ of minimum one-dimensional Hausdorff measure $H$ (length) such that $St \cup \mathcal{A}$ is connected, where $\mathcal{A} \subset \mathbb{R}^d$ is a given compact set. Paolini and Stepanov provided very general existence and regularity results for the Steiner problem. Their main regularity result is that under a natural assumption, $H(St) < \infty$, for almost every $\varepsilon>0$ the set $St_\varepsilon := St\setminus B_\varepsilon(\mathcal A)$ is an embedded finite forest (acyclic graph). We give a quantitative regularity result by proving that the set $St_\varepsilon$ is Ahlfors--David regular with constants that depend only on $d$ (and not on $\mathcal{A}$). Namely, for $d > 2$, every $\varepsilon > 0$, every $x \in St_\varepsilon$, and every choice of $ρ\in (0,1)$, we have \[ \frac{H \left (St_\varepsilon \cap B_{ρ\varepsilon}(x) \right) }{\varepsilon} \leq \left ( \frac{144d}{1-ρ} \right) ^{d-2}. \] As a corollary, we obtain a density-type result, i.e. that the set $St_\varepsilon \cap B_{ρ\varepsilon}(x)$ consists of at most \[ \left ( \frac{144d}{1-ρ} \right) ^{d-1} \] line segments. In the plane (i.e., for $d=2$), it is possible to obtain tight structural results.
Counting the number of spanning trees in specific classes of graphs has attracted increasing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained in the 2020s. The main method is to study the behavior of the vertex (degree) enumerator of a distance-hereditary graph under the operations of copying vertices. Ehrenborg conjecture says that a Ferrer--Young graph maximizes the number of spanning trees among bipartite graphs with the same degree sequence. The second result of this paper is the equivalence of the Ehrenborg conjecture and its polynomial form.
Our previous paper shows that the (vertex) spanning tree degree enumerator polynomial of a connected graph $G$ is a real stable polynomial (id est is non-zero if all variables have positive imaginary parts) if and only if $G$ is distance-hereditary. In this note we generalize the result on weighted graphs. This generalization allows us to define the class of weighted distance-hereditary graphs.
We show that the spanning tree degree enumerator polynomial of a connected graph $G$ is a real stable polynomial if and only if $G$ is distance-hereditary.
Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to $e$ (including $e$ itself) is positive. Then the sum of weights over edges of $G$ is at least $-\frac{n^2}{25}$. Also we provide an example of a weighted graph with described properties and the sum of weights $-(1+o(1))\frac{n^2}{8(1 + \sqrt{2})^2}$. The previous best known bounds were $-\frac{n^2}{16}$ and $-(1+o(1))\frac{n^2}{54}$ respectively. We show that the constant $-1/54$ is optimal under some additional conditions.