# Raziskovalni matematični seminar - Arhiv

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**Datum in ura**/ Date and time: 26.10.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Peter Muršič (UP FAMNIT)

**Naslov**/ Title: Strong cliques in diamond-free graphs.

**Vsebina**/ Abstract:

*A strong clique in a graph is a clique intersecting all inclusion-maximal stable sets. Strong cliques play an important role in the study of perfect graphs. We study strong cliques in the class of diamond-free graphs, from both structural and algorithmic points of view. We show that the following five NP-hard or co-NP-hard problems all remain NP-hard or co-NP-hard when restricted to the class of diamond-free graphs: *

*Is a given clique strong?*

*Does the graph have a strong clique?*

*Is every vertex contained in a strong clique?*

*Given a partition of the vertex set into cliques, is every clique in the partition strong?*

*Can the vertex set be partitioned into strong cliques?*

*On the positive side, we show that the following three problems whose computational complexity is open in general can be solved in polynomial time in the class of diamond-free graphs:*

*Does every induced subgraph have a strong clique?*

*Is every maximal clique strong?*

*Is every edge contained in a strong clique?*

*The last two results are derived from a characterization of diamond-free graphs in which every maximal clique is strong, which also implies an improved Erdos-Hajnal property for such graphs.*

Join Zoom Meeting:

https://upr-si.zoom.us/j/85914318577?pwd=cnFJcmg2ZEZkbFhpeG1VYk0veHp2Zz09

**Datum in ura**/ Date and time: 19.10.20

**Predavalnica**/ Location: FAMNIT-VP1 and Zoom

**Predavatelj**/ Lecturer: Meike Hatzel (TU Berlin, Germany)

**Naslov**/ Title: Constant Congestion Brambles

**Vsebina**/ Abstract:

In this talk I will present a small result we achieved during a workshop in February this year. My coauthors on this are Marcin Pilipczuk, Pawel Komosa and Manuel Sorge.

A bramble in an undirected graph G is a family of connected subgraphs of G such that for every two subgraphs H_1 and H_2 in the bramble either V(H_1) intersects V(H_2) or there is an edge of G with one endpoint in V(H_1) and the second endpoint in V(H_2). The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble.

Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph G equals one plus the treewidth of G. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree n-vertex expander a bramble of order Ω(n^{1/2+δ}) requires size exponential in Ω(n^{2δ}) for any fixed δ \in (0,1/2]. On the other hand, the combination of results of Grohe and Marx, and Chekuri and Chuzhoy shows that a graph of treewidth k admits a bramble of order \widetilde{Ω}(k^{1/2}) and size \widetilde{O}(k^{3/2}). (\widetilde{Ω} and \widetilde{O} hide polylogarithmic factors and divisors, respectively.)

We first sharpen the second bound by proving that every graph G of treewidth at least k contains a bramble of order \widetilde{Ω}(k^{1/2}) and congestion 2, i.e., every vertex of G is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every δ \in (0,1/2], every graph G of treewidth at least k contains a bramble of order \widetilde{Ω}(k^{1/2+δ}) and size 2^{\widetilde{O}(k^{2δ})}.

The seminar will be broadcasting via Zoom through the following link:

Join Zoom Meeting

https://upr-si.zoom.us/j/85914318577?pwd=cnFJcmg2ZEZkbFhpeG1VYk0veHp2Zz09

If you missed it, you can watch the lecture via the following link:

Passcode: K2@ZVpEY

**Datum in ura**/ Date and time: 13.10.20

**Predavalnica**/ Location: FAMNIT-Muzejski4

**Predavatelj**/ Lecturer: Dragan Stevanović (MISANU)

**Naslov**/ Title: On graphs with distinct vertex transmissions

**Vsebina**/ Abstract:

*Transmission of a vertex in a connected graph is the sum of distances from that vertex to all other vertices in the graph. While the use of transmissions as the basis for various topological indices is well known in mathematical chemistry, it is less known that transmissions have been used since 1970s also in space syntax, a particular field of architecture, to denote how easily one can get the sense of a whole building from a locally visible information. Besides these two applications, in this talk we will focus on recent theoretical results on transmissions in graphs, that are mostly related to questions of Klavžar and Dobrynin about graphs in which vertices have distinct transmissions.*

Join Zoom Meeting

https://upr-si.zoom.us/j/85914318577?pwd=cnFJcmg2ZEZkbFhpeG1VYk0veHp2Zz09

**Datum in ura**/ Date and time: 12.10.20

**Predavalnica**/ Location: FAMNIT-VP1 and Zoom

**Predavatelj**/ Lecturer: Isolde Adler (University of Leeds)

**Naslov**/ Title: On the tree-width of even-hole-free graphs

**Vsebina**/ Abstract:

*In recent years, even-hole-free graphs were the object of much attention, however, many questions remain unanswered, such as the existence of a polynomial time algorithm for colouring even-hole-free graphs, or for finding a maximum stable set. The class of even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K4)- free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019]. The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary.*

*We show the class of even-hole-free graphs excluding a minor has bounded tree-width (1). This implies that planar even-hole-free graphs have bounded tree-width [da Silva and Linhares Sales, 2010]. To prove this, we provide an "induced grid theorem" for minor-free graphs.*

*We conjecture that for every d, the class of even-hole-free graphs of degree at most d has bounded tree-width, and we obtain partial results.*

*The class of even-hole-free graphs of degree at most 3 has bounded tree-width.**The class of (even-hole, pyramid)-free graphs of degree at most 4 has bounded tree-width.*

*In the meantime, our conjecture has been proven [Abrishami, Chudnovsky and Vuskovic, 2020].*

*Our second source of motivation for studying the tree-width of even-hole-free graphs stems from the area of property testing, namely, from the question whether even-hole-freeness is testable in the bounded degree model. Indeed, since our conjecture holds, this question is now answered.*

*In this talk I will explain the proof of (1), and give some background on property testing and testability of even-hole-freeness.*

*This is joint work with Pierre Aboulker, Frederik Harwath, Eunjung Kim, Noleen Köhler, Ni Luh Dewi Sintiari and Nicolas Trotignon.*

The seminar will be broadcasting via Zoom through the following link:

Join Zoom Meeting

https://upr-si.zoom.us/j/85914318577?pwd=cnFJcmg2ZEZkbFhpeG1VYk0veHp2Zz09

The slides of the talk are available HERE.

If you missed it, you can watch the lecture via the following link:

Passcode: kKx7cM?n

**Datum in ura**/ Date and time: 5.10.20

**Predavalnica**/ Location: FAMNIT-VP1 and Zoom

**Predavatelj**/ Lecturer: Russ Woodroofe (UP FAMNIT)

**Naslov**/ Title: Combinatorial shifting via limits of matrix actions

**Vsebina**/ Abstract:

Abstract: Combinatorial shifting is a tool used to prove results from extremal set theory such as the Erdős-Ko-Rado theorem. Combinatorial shifting applies a step-by-step procedure to replace a system with a

somewhat simpler (shifted) system.

There are alternate algebraic approaches to shifting. For example, in recent work, I've shown how the Borel Fixed-Point theorem can in certain settings replace (in one step) a system with a shifted system.

In this talk, I'll connect the combinatorial shifting approach with the Borel Fixed-Point approach, by showing how to realize the step-by-step procedure of combinatorial shifting with a limiting

procedure of actions by 1-parameter matrices.

The seminar will be broadcasting via Zoom through the following link:

Join Zoom Meeting

https://upr-si.zoom.us/j/85914318577?pwd=cnFJcmg2ZEZkbFhpeG1VYk0veHp2Zz09

**Datum in ura**/ Date and time: 30.6.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Micael Toledo

**Naslov**/ Title: Cubic vertex-transitive graphs admitting a group of automorphisms with few orbits.

**Vsebina**/ Abstract:

Highly symmetric graphs are frequently studied in terms of the action of a specific type of group of automorphisms, as their structure is often completely determined by this action. Such is the case of a graph G admitting a group of automorphism X that acts transitively on its the arcs, or regularly on its vertices. If X is not vertex-transitive but partitions the vertex-set of G into orbits of equal size, then G can be (and usually is) studied through the theory of covering graphs. However, the case where X partitions the vertex-set of G into few orbits of different sizes is much less understood, and many of the classical tools used in the previously mentioned cases fall short. We give a general overview of the subject, and present new tools and possible lines of research to tackle the problem of classifying cubic graphs admitting a group of automorphisms with a few orbits (possibly of different sizes). We focus particularly on those graphs whose full automorphism group is transitive.

Link for the videoconference:

https://us02web.zoom.us/j/83916090638?pwd=Q3lkanVFZVhRTStnVW9Edi9lRmJhdz09

**Datum in ura**/ Date and time: 15.6.20

**Predavalnica**/ Location: FAMNIT-MP7

**Predavatelj**/ Lecturer: ANDREA MUNARO (Queen's University, Belfast, United Kingdom)

**Naslov**/ Title: Width parameters and graph classes: the case of mim-width

**Vsebina**/ Abstract:

Many computationally hard graph problems can be solved efficiently after placing appropriate restrictions on the input graphs. One reason that might explain this jump in complexity is that the class of restricted inputs has bounded "width". There are several ways of defining a notion of "width", giving rise for example to well-known width parameters such as tree-width and clique-width.

In the talk, I will focus on the width parameter mim-width, introduced by Vatshelle. The modelling power of mim-width is stronger than that of clique-width, in the sense that graphs of bounded clique-width have bounded mim-width but there exist graph classes with bounded mim-width and unbounded clique-width. I will overview structural properties and algorithmic applications of this parameter and present recent joint work with Nick Brettell, Jake Horsfield, Giacomo Paesani and Daniël Paulusma.

The seminar was held by Zoom.

The slides of the talk are available in pdf at this link.

**Datum in ura**/ Date and time: 1.6.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Tomaž Pisanski (UP FAMNIT and UL FMF)

**Naslov**/ Title: Flexible configurations

**Vsebina**/ Abstract:

In a flexible (n_k) configuration at least one configuration line has the property that its k configuration points may be placed on it in k predetermined positions, while rearranging the rest of configuration in such a way that the point-line incidences are preserved.

We will show some useful applications of this concept in connection with Grünbaum incidence calculus. In particular, we will give a proof that for any k and for each sufficiently large number n, there exists an (n_k) configuration of points and lines.

This is joint work in progress with Leah Berman and Gábor Gévay.

Everyone is welcome and encouraged to join the video-lecture through the following link:

https://us02web.zoom.us/j/

**Datum in ura**/ Date and time: 25.5.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Andrés David Santamaría-Galvis (UP FAMNIT)

**Naslov**/ Title: PARTITIONING THE PROJECTIVE PLANE AND THE DUNCE HAT

**Vsebina**/ Abstract:

**Datum in ura**/ Date and time: 18.5.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Žiga Velkavrh (UP FAMNIT and UP IAM)

**Naslov**/ Title: Market makers, information, and bid-ask spread

**Vsebina**/ Abstract:

The talk starts with the brief introduction of Mechanism Design theory, with particular focus on dominant strategy incentive compatible mechanisms. Next, the joint work with Prof. Dr. J. Čopič and Dr. M. Castellani is presented. We study the bilateral trade problem under no subsidies constraint which has not been explored much due to its complexity. In our setting, there are two traders, i.e., a seller and a buyer, one indivisible good, and a market maker. The traders have private valuations over one indivisible good and have a scarce information. A market maker is in full command of the choice of market platform and has no private information. We consider market equilibrium mechanisms, among which a special attention is devoted to stochastic bid-ask mechanisms. We conclude with an example that shows that in general a mechanism that maximizes a social welfare function may lie outside the set of simple posted prices mechanisms.

Link Zoom: https://us02web.zoom.us/j/86227923652

**Datum in ura**/ Date and time: 11.5.20

**Predavalnica**/ Location: ZOOM

**Predavatelj**/ Lecturer: Blas Fernandez (UP FAMNIT - UP IAM)

**Naslov**/ Title: On the Terwilliger algebra of distance-regularised graphs

**Vsebina**/ Abstract:

Terwilliger algebra of a commutative association scheme was successfully used for studying distance-regular graphs. However, this notion can be easily generalized to an arbitrary finite, simple and connected graph. In many instances, this algebra can be better studied via its irreducible modules. In this talk, we show how to relate the module structure of the Terwilliger algebra of a graph with the concept of local distance-regularity. Finally, we investigate how this algebra can be used as an effective tool for studying some combinatorial properties in the graph.

This is joint work with Stefko Miklavic.

Nut graphs are singular graphs of nullity one with full corresponding eigenvector. Recently, the existence problem of regular nut graphs was successfully addressed. In this talk, we present a generalisation of nut graphs to signed nut graphs and give some preliminary results of their existence. In particular, Fowler construction, switching equivalence and connection to double covers will be presented.

This is work in progress with Patrick W. Fowler, Irene Sciricha and Nino Bašić.

Link for the Conference: https://us02web.zoom.us/j/87925579576

**Datum in ura**/ Date and time: 6.4.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Vladimir BATAGELJ (IMFm, UP IAM, Slovenia and NRU HSE, Russia)

**Naslov**/ Title: NetsJSON

**Vsebina**/ Abstract:

A network consists of a graph and additional data on (properties of) nodes and/or links. In some applications, the values of properties can be structured objects such us intervals, compositions, sets of keywords, temporal quantities, etc.

Standard network description formats do not support such kinds of networks. We propose a JSON based format NetsJSON as a general network description option. It is supported in Python by a network analysis library Nets. The NetsJSON descriptions can be used as input data for network visualizations based on D3.js.

REFERENCES:

[1] Batagelj, V.: Python Packages for Networks. Encyclopedia of Social Network Analysis and

Mining. Reda Alhajj, Jon Rokne (Eds.), Springer, 2017

https://link.springer.com/referenceworkentry/10.1007%2F978-1-4614-7163-9_110210-1

[2] Batagelj, V., Maltseva, D.: Temporal bibliographic networks. Journal of Informetrics,

14(2020)1, 101006. https://www.sciencedirect.com/science/article/pii/S1751157719301439

[3] Batagelj, V.: Nets. https://github.com/bavla/Nets

[4] Batagelj, V.: NetsJSON. https://github.com/bavla/netsJSON

**Datum in ura**/ Date and time: 30.3.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: András BÓTA

**Naslov**/ Title: NETWORK-BASED FRAMEWORK FOR ESTIMATING THE RISK FACTORS OF EPIDEMIC SPREADING

**Vsebina**/ Abstract:

Link Zoom: https://zoom.us/j/297328207

Network-based adaptations of traditional compartmental infection models such as SIR or SEIR can be used to model the spreading of diseases between cities, countries or other geographical regions. One of the common challenges arising in these applications is the lack of available transmission probabilities between these geographical units. The task of inverse infection is the systematic estimation of these values. Several methods have been proposed recently for solving this task. One of them is the Generalized Inverse Infection Model (GIIM) [1]. GIIM offers a large amount of modeling flexibility and allows transmission probabilities to be defined as a function of known attributes, or risk factors in an epidemic context. In this presentation we will see how GIIM works in two specific real-life outbreaks.

Both examples are embedded in a geographical and temporal setting. The first one considers the 2015-2016 Zika virus outbreak in the Americas, where the countries and overseas territories of the continent form the nodes of the network and air travel routes define the links [2]. The second application models the 2009 H1N1 outbreak between the municipalities of Sweden, with links between the municipalities indicating frequent travel routes.

Our first goal in both of these studies is to discover the relationship between the transmission risk between geographical units and a variety of travel, environmental, meteorological and socioeconomic risk factors. Our second goal is to estimate the risk of exportation and importation of the diseases for the territories involved in these studies. We will show that the GIIM model is able to identify the most critical risk factors in both scenarios, and in the influenza study, it is able make predictions about future outbreaks with good accuracy.

REFERENCE:

1. A. Bóta, L. M. Gardner: A generalized framework for the estimation of edge infection probabilities. arXiv:1706.07532 (2017)

2. L. M. Gardner, A. Bóta, N. D. Grubaugh, K. Gangavarapu, M. U. G. Kramer: Inferring the risk factors behind the geographical spread and transmission of Zika in the Americas.PLoS Neglected Tropical Diseases.

http://journals.plos.org/plosntds/article?id=10.1371/journal.pntd.0006194

**Datum in ura**/ Date and time: 23.3.20

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Tomaž Pisanski (UP FAMNIT and UL FMF)

**Naslov**/ Title: Why is distance learning tough?

**Vsebina**/ Abstract:

You should understand…

- I am talking not as an expert but as a long-time university lecturer who likes distance teaching and learning.
- Due to unusual circumstances, we are all pushed into distance learning and distance teaching. This serves as a good excuse to talk about such unusual topic at mathematics seminar.
- Throughout our University there are many disscussions about choosing appropriate tools for teaching courses via internet.
- We can imagine that such discussions take place in many other places in Slovenia and elsewhere on Earth.

I am not going to…

- ... promote one system over another one. [I hope some of you, will be able to do it soon at a similar seminar here or at some other location.]
- ... teach you how to use Powerpoint or Beamer, etc. for preparing your Seminar Talk.
- ... teach you how to organize your courses in moodle.
- ... teach you how to make and upload videos to Youtube on Vimeo.

So what I am going to do today?

- I will try to sketch a “theory” of communcation.
- Again, since I am not an expert in this area the word “theory” has to be taken
**cum grano salis.** - The goal of this talk is to give you enough theory that will explain
- why distance learning is difficult.
- why teaching large groups of students is more complicated than teaching small groups.
- why overusing colors and animation may be counter.

Everyone is welcome and encouraged to join the video-lecture via the following link:

https://zoom.us/j/479411024

**Datum in ura**/ Date and time: 9.3.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Daniel Hawtin (University of Rijeka, Croatia )

**Naslov**/ Title: 2-Neighbour-Transitive Codes and Completely Transitive Codes

**Vsebina**/ Abstract:

A code is a subset of the vertex set of a graph. Given a code, the graph metric allows one to define an associated distance partition. Imposing combinatorial regularity conditions on the distance partition of a code leads to the definitions of the classes of s-regular and completely regular codes; analogously, algebraic symmetry conditions lead to the classes of s-neighbour-transitive and completely transitive codes. I will discuss previous results and current work related to characterising and classifying subclasses of 2-neighbour-transitive and completely transitive codes in Hamming graphs. All of the results I will discuss are part of ongoing effort to provide a full classification of completely transitive codes in Hamming graphs having minimum distance at least 5.

**Datum in ura**/ Date and time: 2.3.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Nejc Šarabon (Faculty of Health Sciences, UP )

**Naslov**/ Title: MULTI-LAYER INSIGHT INTO BODY ASYMMETRIES IN THE CONTEXT OF SPORTS INJURIES PREVENTION AND CARE

**Vsebina**/ Abstract:

The BOSY project is addressing body (a)symmetries of athletes in different sport disciplines, including soccer, basketball, tennis, running and more. Local (single-joint or single-body part) and global asymmetries in muscle strength, power, flexibility and body stability are a very frequent phenomenon in athletes. Despite the increasing knowledge in the field of prevention and treatment, and represent one of the risk factors for sustaining an injury. Injuries have a detrimental effect both on an individual level, as athlete may be required to abstain from sport participation for significant amount of time, and on the level of clubs, sport associations and society, as injuries are often expensive to treat. Researchers within this project wish to contribute to the much needed knowledge on associations between (a)symmetries and sports injuries, and then further explore different measures for eliminating asymmetries and treating sports injuries.

Within the project, we are recruiting a large pool of athletes and carry out several test on them to seek for potential (a)symmetries. We are also retrospectively and prospectively examining the associations between the recognized asymmetries and occurrence of injuries. The test battery includes several aspects of physical ability and function (strength, power, balance, flexibility and kinematics of cyclic and acyclic movement patterns). The exceptionally large and rich dataset will enable to assess relationships, group differences, within-subject differences and other more complex calculations. With the aid of the experts of statistics and mathematics, the analyses could be stepped up to investigate potential patterns in the data that are not discoverable with common data processing methods. Additional insights, arising from such computations could reveal different, possibly more important and practically relevant information in terms of understating the underlying mechanisms of asymmetries and designing approaches to prevent injuries in sport.

**Datum in ura**/ Date and time: 24.2.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Russ Woodroofe (UP FAMNIT)

**Naslov**/ Title: A New Characterization of Supersolvable Lattices

**Vsebina**/ Abstract:

Stanley introduced _supersolvable lattices_ in the 1970s to describe properties that certain lattices have in common with subgroup lattices of supersolvable groups. In recent work with Stephan Foldes, we have given a new characterization of supersolvable lattices in terms of lattice-theoretic analogues of normal subgroups. Our characterization simplifies and unifies existing literature on this class.

I'll introduce this class of lattices with motivating examples, and compare our characterization with existing ones.

**Datum in ura**/ Date and time: 15.2.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Stevan Pilipović & Gradimir V. Milovanović

**Naslov**/ Title: Mathematical Research Seminar sponsored by SASA at UP FAMNIT

**Vsebina**/ Abstract:

10:00 – 10:45: FAMNIT-MP1__Lecturer: __Stevan Pilipović__Title: __*Pseudo-differential operators-some new results.*

Stevan Pilipović is an academician of the Serbian Academy of Sciences and Arts. His research interests include functional analysis, generalized functions and hyperfunctions, pseudo-differential operators, time-frequency analysis, linear and nonlinear equations with singularities, probability theory and stochastic processes.

**11:00 – 11:45: FAMNIT-MP1**__Lecturer:__ Gradimir V. Milovanović __Title: __*Orthogonality in the complex plane and some applications.*

Gradimir V. Milovanović is a full member of the Serbian Academy of Sciences and Arts. His research is motivated by the range of topics among orthogonal polynomials and systems, interpolation theory and numerical analysis.

**Datum in ura**/ Date and time: 10.2.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Gábor Korchmáros (Universitá degli Studi della Basilicata, Potenza, Italy)

**Naslov**/ Title: Problems in Euclidean Distance Geometry

**Vsebina**/ Abstract:

Euclidean distance geometry, that is the study of Euclidean geometry based on the concept of distance, is of current interest in several practical applications, such as molecular biology, wireless sensor networks, statics, data visualization and robotics. In this talk, we show how introductory algebraic geometry can be used as an effective tool for the solution of certain problems in Euclidean distance geometry.

**Datum in ura**/ Date and time: 27.1.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Meike Hatzel (TU Berlin, Germany)

**Naslov**/ Title: Avoidable paths in graphs

**Vsebina**/ Abstract:

**Datum in ura**/ Date and time: 13.1.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Gyula Y. Katona (Budapest University of Technology and Economics, Hungary)

**Naslov**/ Title: Minimally t-Tough Graphs

**Vsebina**/ Abstract:

A graph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases the toughness. Kriesell conjectured that for every minimally1-tough graph the minimum degree δ(G) = 2. It is natural to generalize this for other t values: Every minimally t-tough graph has a vertex of degree ceil(2t). In the present talk we investigate different questions related to this conjecture. The conjecture seems to be hard to prove, so we tried to prove it for some special graph classes. It turned out, that in some cases the conjecture is true because there are very few graphs that satisfy the conditions. On the other hand, we have evidence using complexity theory, that this is not the situation for some other graph classes. Many open questions remain.

This is joint work with Kitti Varga, István Kovács, Dániel Soltész.

**Datum in ura**/ Date and time: 6.1.20

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Nino Bašić (UP FAMNIT)

**Naslov**/ Title: Point-ellipse and some other exotic configurations

**Vsebina**/ Abstract:

In this talk, we introduce point-ellipse configurations and point-conic configurations. We present some of their basic properties and describe two interesting families of balanced point-conic 6-configurations. The construction of the first family is based on Carnot's theorem, whilst the construction of the second family is based on the Cartesian product of two regular polygons. Finally, we investigate a point-ellipse configuration based on the regular 24-cell.

This is joint work with Gábor Gévay, Jurij Kovič and Tomaž Pisanski.