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Andrei Pavelescu

Publications and source records attributed to Andrei Pavelescu.

16 recordsLinked to original sources
arXiv2026

Three thousand obstructions to knotless embedding

We present a list of 3028 obstructions to knotless embedding. We survey recent work in this area including: 1) A bibliography of graphs proven to be intrinsically knotted without relying on computers; 2) An updated listing of obstructions in $\nabla\mathrm{Y}$ families including two new large families; 3) Connections with the Colin de Verdière's invariant including a new obstruction with $μ= 6$; and 4) Connectivity of obstructions and their structure near vertices of degree three or four. We address questions raised in earlier work, (re)state several conjectures, and propose new questions.

math.GT↗
arXiv2026

New minor minimal non-apex graphs

A graph is apex if it becomes planar after the deletion of one vertex. The family of apex graphs is closed under taking minors, so it is characterized by a finite set of forbidden minors. Determining the finite set of forbidden minors for apex graphs remains an open question. In this paper, we list all forbidden minors for apex graphs with 12 or fewer vertices and all forbidden minors for apex graphs with 26 and fewer edges. We also present graphs outside of these ranges. We show that a graph with 13 vertices and minimal degree 6 is either apex or contains a $K_6$ minor, proving Jørgensen's conjecture for order 13.

math.CO↗
arXiv2024

Intrinsically knotted graphs and connected domination

We classify all the maximal linklessly embeddable graphs of order 12 and show that their complements are all intrinsically knotted. We derive results about the connected domination numbers of a graph and its complement. We provide an answer to an open question about the minimal order of a 3-non-compliant graph. We prove that the complements of knotlessly embeddable graphs of order at least 15 are all intrinsically knotted. We provide results on general $k$-non-compliant graphs and leave a set of open questions for further exploration of the subject.

math.CO↗
arXiv2022

The Complement Problem for Linklessly Embeddable Graphs

We find all maximal linklessly embeddable graphs of order up to 11, and verify that for every graph $G$ of order 11 either $G$ or its complement $cG$ is intrinsically linked. We give an example of a graph $G$ of order 11 such that both $G$ and $cG$ are $K_6$-minor free. We provide minimal order examples of maximal linklessly embeddable graphs that are not triangular or not 3-connected. We prove a Nordhaus-Gaddum type conjecture on the Colin de Verdière invariant for graphs on at most 11 vertices. We give a description of the programs used in the search.

math.GT↗
arXiv2022

Complete Minors in Complements of Non-Separating Planar Graphs

We prove that the complement of any non-separating planar graph of order $2n-3$ contains a $K_n$ minor, and argue that the order $2n-3$ is lowest possible with this property. To illustrate the necessity of the non-separating hypothesis, we give an example of a planar graph of order 11 whose complement does not contain a $K_7$ minor. We argue that the complements of planar graphs of order 11 are intrinsically knotted. We compute the Hadwiger numbers of complements of wheel graphs.

math.CO↗
arXiv2022

Constructions stemming from non-separating planar graphs and their Colin de Verdière invariant

A planar graph $G$ is said to be non-separating if there exists an embedding of $G$ in $\mathbb{R}^2$ such that for any cycle $\mathcal{C}\subset G$, all vertices of $G\setminus \mathcal{C}$ are within the same connected component of $\mathbb{R}^2\setminus \mathcal{C}$. Dehkordi and Farr classified the non-separating planar graphs as either outerplanar graphs, subgraphs of wheel graphs, or subgraphs of elongated triangular prisms. We use maximal non-separating planar graphs to construct examples of maximal linkless graphs and maximal knotless graphs. We show that for a maximal non-separating planar graph $G$ with $n\ge 7$ vertices, the complement $cG$ is $(n-7)-$apex. This implies that the Colin de Verdière invariant of the complement $cG$ satisfies $μ(cG) \le n-4$. We show this to be an equality. As a consequence, the conjecture of Kotlov, Lovàsz, and Vempala that for a simple graph $G$, $μ(G)+μ(cG)\ge n-2$ is true for 2-apex graphs $G$ for which $G-\{u,v\}$ is planar non-separating. It also follows that complements of non-separating planar graphs of order at least nine are intrinsically linked. We prove that the complements of non-separating planar graphs $G$ of order at least ten are intrinsically knotted.

math.CO↗
arXiv2020

New bounds on maximal linkless graphs

We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly improves the lowest edge-to-vertex ratio for any previously known infinite family. We construct a family of graphs showing that the class of maximal linklessly embeddable graphs differs from the class of graphs that are maximal without a $K_6$ minor studied by L. Jorgensen. We give necessary and sufficient conditions for when the clique sum of two maximal linklessly embeddable graphs over $K_2$, $K_3$, or $K_4$ is a maximal linklessly embeddable graph, and use these results to prove our constructions yield maximal linklessly embeddable graphs.

math.GT↗
arXiv2018

Complete Minors of Self-Complementary Graphs

We show that any self-complementary graph with $n$ vertices contains a $K_{\lfloor \frac{n+1}{2}\rfloor}$ minor. We derive topological properties of self-complementary graphs.

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arXiv2018

Hadwiger numbers of self-complementary graphs

The Hadwiger number of a graph $G$, denoted by $h(G)$, is the order of the largest complete minor of $G$. A graph is said to be self-complementary if it is isomorphic to its complement. We prove that for all $n\equiv 0,1 (\text{mod 4})$ and for all $ \lfloor \dfrac{n+1}{2} \rfloor \le h \le \lfloor \dfrac{3n}{5}\rfloor $, there exists a self-complementary graph $G$ with $n$ vertices whose Hadwiger number is $h$.

math.CO↗
arXiv2018

Escher squares and lattice links

We give a shorter and simpler proof of the result of [2], which gives a necessary and sufficient condition for when a lattice diagram is the projection of a lattice link.

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arXiv2013

Derangements in Cosets of Primitive Permutation Groups

Motivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher-O'Nan-Scott theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.

math.GR↗