Mirka Miller's Combinatorics Webinar Series

Welcome to the home page for Mirka Miller's Combinatorics Webinar Series.

Meeting link: https://meet.google.com/kgc-uwpc-ngp

Upcoming Talks

Date: Wednesday September 15 2021

Time: 10:00 (CET)

Speaker: Prof. Robert Jajcay

Title: Biregular Cages

Abstract: The Cage Problem - the problem of finding a smallest k-regular graph of girth g, i.e., the (k,g)-cage - is well known to be very hard and the exact orders of cages are known for very few parameter pairs (k,g). One possible approach to understanding structural properties of cages includes considering biregular graphs that contain vertices of two degrees, m and n, and generalizing the Cage Problem by looking for smallest graphs of girth g containing vertices of the two degrees m and n, the (m,n;g)-cages. In the case of odd girths, results of this approach differ quite a bit from the regular Cage Problem as the orders of biregular (m,n;g)-cages are determined for all odd girths g and degree pairs m,n in which m is considerably smaller than n. The even girth case is still wide open, and has been therefore restricted to bipartite biregular graphs in which the two bipartite sets consist exclusively of vertices of one of the degrees (regular cages of even girth are also conjectured to be bipartite). We survey the most resent results on biregular and bipartite biregular cages, present some improved lower bounds, and discuss an interesting connection between bipartite biregular cages and t-designs.

Previous Talks

Talk 1: Celebrating Mirka's life - Prof. Camino Balbuena

Date: March 8th 2021

Time 17:00 (CET)

Celebrating Mirka's life

Chair: Prof. Gabriela Araujo-Pardo

"Remembering Prof. Mirka Miller"

by Prof. Cristina Dalfó

Speaker: Prof. Camino Balbuena

Title: On the Moore cages with a prescribed girth pair

Abstract: https://drive.google.com/file/d/11r-TY8NswuKZkT7LOfkl-4yy1d72aF-w/view?usp=sharing

Slides: https://drive.google.com/file/d/1GkbwwNYSLBgRGjE8z7n1ckuXbXb6sRrN/view?usp=sharing

Video: https://drive.google.com/file/d/1DuWvsAsxMoIzCH4wr_hpGqUCx5uv89SU/view?usp=sharing {Chair: Prof. Gabriela Araujo-Pardo (minute 0:00-04:08) - "Remembering Prof. Mirka Miller" by Prof. Cristina Dalfó (minute 04:08-17:44) - Speaker: Prof. Camino Balbuena On the Moore cages with a prescribed girth pair (minute 17:44-end)}

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Talk 2: May there be many more repeats - Prof. Jozef Širáň

Date: Wednesday April 14 2021

Time: 1000 Bratislava (0900 UK)

Speaker: Prof. Jozef Širáň

Title: May there be many more repeats

Abstract: This is my reminiscence on two mathematical aspects of my collaboration with Mirka Miller in the degree-diameter problem: the lifting technique in constructions of `large' examples and her method of repeats in non-existence proofs.

Video: https://drive.google.com/file/d/1Ss2xvDP9UAIebNUnIr2U6J01wixUJFkK/view?usp=sharing {Chair: Prof. Camino Balbuena (minute 0:00-02:26) - Speaker: Prof. Jozef Širáň May there be many more repeats (minute 02:26-end)}

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Talk 3: The Domination Blocking Game - Prof. Dominique Buset

Date: Wednesday May 12 2021

Time: 11:00 (CET) - one hour later than the previous one

Speaker: Prof. Dominique Buset

Title: The Domination Blocking Game

Abstract: We introduce a new game on a simple, finite and undirected graph: “the domination tracking game”. Two players (the Dominator and the Enemy), each one playing alternatively, take a not occupied vertex on the graph. When the dominator (resp. the enemy) takes a vertex, he controls the vertex and all its neigbours (resp. just the vertex taken). The purpose of the game is for the dominator to control all the vertices, and for the enemy to avoid the dominator to win (i.e. to take one vertex and all his neighbours). We determine for some categories of graphs a winning strategy either for the Dominator or the Enemy. These situations, give a partition of those graphs into three classes. https://drive.google.com/file/d/1V8WY_qlqCDHdZLFsTokOQ-7t7KBtnqSG/view?usp=sharing

As part of the initiatives of Women in Mathematics Day

Video: https://drive.google.com/file/d/1HIv1fixjrWZfcRYHwN3QgntBKMoMK1Mt/view?usp=sharing {Chair: Prof. Cristina Dalfó - Speaker: Prof. Dominique Buset The Domination Blocking Game }

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Talk 4: The story about graphs CD(k,q) - Prof. Felix Lazebnik

Date: Wednesday June 16 2021

Time: 18:00 (CET)

Speaker: Prof. Felix Lazebnik

Title: The story about graphs CD(k,q)

Abstract: In this talk I will present the main ideas and history behind the construction of the family of graphs that is usually denoted by CD(k,q), where k is a positive integer, and q is a prime power. It is known that the girth of CD(k,q) (the length of its shortest cycle) is at least k+5, and these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of a given order and given girth at least g, where g ≥ 5 and g distinct from 11, 12. We survey some old and new results, and mention several open questions related to these graphs or to similarly constructed graphs.

Slides: https://drive.google.com/file/d/13G9bVkwBEXgA6bkX4MJOgoAJsFRNorMs/view?usp=sharing

YouTube version of Cohen’s `Anthem’: https://youtu.be/c8-BT6y_wYg

Lev Arkad'evich Kaluznin (with M.H. Klin, G. Pöschel, V.I. Suschansky, V.A. Ustimenko, V.I. Vyshensky), Applicandae Matematicae, 52:(1998) 5--18 MR 99m: 01091. https://drive.google.com/file/d/1Ak_MAzyE2FEW6kEflHi_1hDMghOiVp48/view?usp=sharing

General properties of some families of graphs defined by systems of equations. (joint work with A.J. Woldar), Journal of Graph Theory, 38, (2001), 65--86. MR 2002k: 05108. https://drive.google.com/file/d/1vJCV9CXeOyfCoGp0Epmatw2n4BLu9be3/view?usp=sharing

Some Families of Graphs, Hypergraphs and Digraphs Defined by Systems of Equations: A Survey. (with Shuying Sun and Ye Wang), Lecture Notes of Seminario Interdisciplinare di Matematica , Vol. 14 (2017), pp. 105–-142. https://drive.google.com/file/d/13HV9UrloLrxGt62_YOOJ-WT4cgDOZAiN/view?usp=sharing

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Talk 5: A Journey with Antimagic Labeling - Prof. Kiki Ariyanti Sugeng

Date: Wednesday July 14 2021

Time: 15:00 (CEST)

Speaker: Prof. Kiki Ariyanti Sugeng

Title: A Journey with Antimagic Labeling

Abstract: Antimagic labeling is defined as an assignment from the element of a graph to usually a set of integers such that the weight of the element of the graph is all different. There are many variations of antimagic labeling, depending on which element of graph is labeled and how the weight is calculated. One of the definitions is as follows: A graph G is called antimagic if the edges can be labeled with the integers 1,2,...,q such that the sum of labels at any given vertex is different from the sum of the labels at any other vertex, i.e., no two vertices have the same sum. In this talk, I would like to share my journey with antimagic labeling through many variations of this labeling. https://drive.google.com/file/d/11AXXVAbNeJ9LbskDD_jZNoC9O2-XLHkP/view?usp=sharing

Slides: https://drive.google.com/file/d/1HGHz4e7pV6GJ1kICdbvVt8racCiMeZoN/view?usp=sharing

Video: https://drive.google.com/file/d/1YM18ibCuuB1XYH8A8XkK_S_I7yY_cCNg/view?usp=sharing

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Mirka's Research and related works

Obituaries: Mirka Miller (nee Koutova) https://drive.google.com/file/d/1su6y1qhUkR3PosqRDfFSGhFbKz1oseP1/view?usp=sharing

Special Issue in Honour of Mirka Miller https://drive.google.com/file/d/1wWRIXelvnGhkOM7IlhZhH0G0xxT6rd0s/view?usp=sharing

In memoriam Emeritus Professor Mirka Miller https://drive.google.com/file/d/1WQVHk41Yi5fJuGhWYscN9CwwirdU8uOu/view?usp=sharing

Eulogy for Professor Mirka Miller (1949–2016) [v https://drive.google.com/file/d/1hTS8HxU9omjlF2Q-_jOQIREcguR1Y1FU/view?usp=sharing]

A family of mixed graphs with large order and diameter 2 https://drive.google.com/file/d/1zbwb56QKY1iSeOm6c4ko178Y0UXNIzfL/view?usp=sharing

Photos

Organisers

• Marién Abreu - Dipartimento di Matematica, Informatica ed Economia - Università degli Studi della Basilicata - Potenza, Italia

• Gabriela Araujo-Pardo - Mathematics Institute-Juriquilla - Universidad Nacional Autónoma de México, México

• Camino Balbuena - Department of Civil and Enviromental Engineering - Universitat Politècnica de Catalunya, Spain

• Cristina Dalfó - Cryptography and Graphs Research Group - Universitat de Lleida, Igualada (Barcelona), Catalonia

• Tatiana Jajcayova - Faculty of Mathematics, Physics and Informatics - Comenius University, Bratislava, Slovakia

Honorary Organiser

• Joe Ryan - School of Electrical Engineering and Computing - University of Newcastle, Australia

Website Host and Support

• Grahame Erskine - Department of Mathematics of Statistics - The Open University, UK.

(the organisers thank him for kindly including this website in Combinatorics Wiki)