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 March 15, 2023

Time: TBA

Speaker: Prof. Clemens Huemer (UPC)

Title: TBA

Abstract: TBA

---

Previous Talks

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

Date: March 8 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)}

---

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)}

---

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 }

---

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

---

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

---

Talk 6: Biregular Cages - Prof. Robert Jajcay

Date: Wednesday September 15 2021

Time: 16:00 (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.

Slides: https://drive.google.com/file/d/1GKlTVWu63BaPMhbEvPOibC8_CeQki3iK/view?usp=sharing

Video: https://drive.google.com/file/d/1cFV_jLyoge-NvFLGakkyxU-jhzWOq5s0/view?usp=sharing

---

Talk 7: Open problema in Distance-based Graph Labelings - Prof. Rinovia Simanjuntak

Date: Wednesday October 13 2021

Time: 13:00 (CET)

Speaker: Prof. Rinovia Simanjuntak

Title: Open problema in Distance-based Graph Labelings

Abstract: https://drive.google.com/file/d/1zLaEyrimiZphyQ-wWDMvi_SW8JS-uu59/view?usp=drivesdk

Slides: https://drive.google.com/file/d/1RW19I1fUWR0d7lHd13xQb1oHQbVVCYy6/view?usp=sharing

Video: https://drive.google.com/file/d/1FrUpH3vgGCOwzvTGuRnZ9uqH4CPSfbxF/view?usp=sharing

---

Talk 8: On the existence of almost Moore digraphs - Prof. Edy Tri Baskoro

Date: Wednesday November 10 2021

Time: 15:00 (CET)

Speaker: Prof. Edy Tri Baskoro

Title: On the existence of almost Moore digraphs

Abstract: https://drive.google.com/file/d/1L6t25SxIUEJAyt2z1F5jdLSFjgVbUfBv/view?usp=sharing

Video https://drive.google.com/file/d/1uDJtL9Anut5n2XT5fQA6PsXC1wOod-kH/view?usp=sharing

---

Talk 9: Strong cliques in diamond-free graphs - Prof. Berenice Martinez Barahona

Date: Wednesday December 15 2021

Time: 17:00 (CET)

Speaker: Prof. Berenice Martínez-Barona

Title: Strong cliques in diamond-free graphs

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. In this talk, 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 two problems whose computational complexity is open in general can be solved in linear 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 Erdös-Hajnal property for such graphs.

(Joint work with Nina Chiarelli, Martin Milanič, Jerome Monnot and Peter Muršič.)

Video https://drive.google.com/file/d/10rZ2fGwCdjR_uZvIqDuLIG5CXcTRdt1m/view?usp=sharing

---

Talk 10: Perfect Matchings in Regular Graph with Given Connectivity - Prof. Yuqing Lin

Date: Wednesday January 19 2022

Time: 9:30 Sydney time [23:30 (CET) of January 18th 2022]

Speaker: Prof. Yuqing Lin

Title: Perfect Matchings in Regular Graph with Given Connectivity

Abstract: The problem of how many edge disjoint perfect matchings are there in a graph is a classical topic in graph theory. Most of the work focus on the case where the degree is large, roughly speaking, equal to half of the total number of the vertices in the graph. For example, it has been conjectured that these graphs can be factorized. When working with this conjecture, we realize that the connectivity plays an important role in guarantee the graph has required number of perfect matchings and/or other factors. In this talk, I will present a few results we have obtained along this direction.

Video https://drive.google.com/file/d/1XIHFQKK8cPCccgEuNU8LEchtovKZfH7p/view?usp=sharing

---

Talk 11: Colorings of Moore graphs - Prof. Mika Olsen

Date: Wednesday February 16 2022

Time: 18:00 (CET)

Speaker: Prof. Mika Olsen - Universidad Autónoma Metropolitana- Cuajimalpa (UAM-Cuajimalpa), México

Title: Colorings of Moore graphs

Abstract: Coloring graphs is a popular topic in graph theory. In this talk I consider two different colorings of Moore graphs, namely rainbow colorings of edges and packing colorings of vertices.

A path P in an edge-colored graph is a rainbow path if there are no arcs of P with the same color. The rainbow connectivity rc(G) is the smallest integer k such that there is an edge coloring of a graph satisfying that the graph is connected by rainbow paths.

A vertex coloring of a graph is a packing coloring if any pair of vertices of color i satisfies that the distance between them is at least i+1. The packing chromatic number is the is the smallest integer k such that there is a packing coloring with k colors.

I will present results for the rainbow t-connectivity of (k,6)-Moore graphs and results for packing chromatic number of Moore graphs with girth 6,8 and 12. In both cases the results where obtained using structures of Moore graphs.

Video https://drive.google.com/file/d/13kyGioVoVGcKliUAt_0afyEJmgdtBiZK/view?usp=sharing

---

Talk 12: Four families of polynomials in spectral graph theory - Prof. Miquel Àngel Fiol

Date: Wednesday March 16 2022

Time: 17:00 (CET)

Speaker: Prof. Miquel Àngel Fiol

Title: Four families of polynomials in spectral graph theory

Abstract: In this talk, we describe four families of polynomials related to the spectrum of a graph. First, some known main results involving such polynomials, such as the spectral excess theorem characterizing distance-regularity, are discussed. Second, some new results giving bounds for the k-independence number αk of a graph are presented. In this context, we comment on some relationships between the inertia (Cvetkovic) and ratio (Hoffman) bounds of αk.

Slides: [https://drive.google.com/file/d/1EbfdMPDFXfs1E8CtUbCBtrSw7mT-gE2f/view?usp=sharing https://drive.google.com/file/d/1EbfdMPDFXfs1E8CtUbCBtrSw7mT-gE2f/view?usp=sharing]

Video: [https://drive.google.com/file/d/1MyQ-SsQ5bXVc4nUYmyjxGCSwVaH3iNQo/view?usp=sharing https://drive.google.com/file/d/1MyQ-SsQ5bXVc4nUYmyjxGCSwVaH3iNQo/view?usp=sharing]

---

Talk 13: Berge’s conjecture for cubic graphs with small colouring defect - Prof. Edita Máčajová

Date: Wednesday April 20 2022

Time: TBA

Speaker: Prof. Edita Máčajová

Title: Berge’s conjecture for cubic graphs with small colouring defect

Abstract: https://drive.google.com/file/d/1tAEybLG2Pg1G6DYQSaTW14y1HOqteIFa/view?usp=sharing

Video: https://drive.google.com/file/d/1kgAmb7zQB9Jvsrr4y8Lt0pgK1JB4z-L6/view?usp=sharing

---

Talk 14: Measuring the closeness to Moore graphs - Prof. Nacho López

Date: Wednesday May 18 2022

Time: 15:00 (CET)

Speaker: Prof. Nacho López

Title: Measuring the closeness to Moore graphs

Abstract: In this talk we will discuss how to measure the closeness to a Moore graph. This `closeness' has been usually measured as the difference between the (unattainable) Moore bound and the order of the considered graphs. Another kind of approach considers relaxing some of the constraints implied by the Moore bound. For instance, we could relax the condition of the degree and admit few vertices with larger degree, as Mirka did for the directed case. Radial Moore graphs are also approximations of Moore graphs, where we allow the existence of vertices with eccentricity just on more than the value they have in a Moore graph. Once we have many different graphs close to Moore graph, an interesting question is how to rank them according to their proximity to being a Moore graph.

Slides: https://drive.google.com/file/d/1FeqbpnXDWpS7xd37Zknyh3ZQ15H0JR38/view?usp=sharing

Video: https://drive.google.com/file/d/1C7zLSb3b_anh3oXE2MZz0BpoPXVOZN_6/view?usp=sharing

---

Talk 15: On mixed cages - Prof. Diego González

Date: Wednesday June 15 2022

Time: 17:00 (CET)

Speaker: Prof. Diego González

Title: On mixed cages

Abstract: A mixed graph is a graph with edges and arcs. A [z, r; g]-mixed cage is a mixed graph, z-regular by arcs, r-regular by edges with girth g and minimum order. In this talk I’m going to give an overview of the results in mixed cages, and I will talk about the monotonicity and the connectivity of mixed cages. This is a joint work with Gabriela Araujo and Claudia De la Cruz.

Slides: TBA

Video: TBA

---

Talk 16: On weight-equitable partitions of graphs - Prof. Aida Abiad

Date: Wednesday July 20 2022

Time: 14:00 (CET)

Speaker: Prof. Aida Abiad

Title: On weight-equitable partitions of graphs

Abstract: Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several classic eigenvalue bounds. Weight-equitable partitions assign to each vertex a weight that equals the corresponding entry of the Perron eigenvector. In this talk we will present several algebraic characterizations and some computational results of weight-equitable partitions. In addition, we will show that such partitions provide a condition under which Hoffman's ratio bound can be improved.

Slides: TBA

Video: https://drive.google.com/file/d/1Desep_26gt_gg7W7-M8VrGg_uTdrB-x_/view?usp=sharing

---

Talk 17: Spectral Moore theorems for graphs and hypergraphs - Prof. Sebastian Cioaba

Date: Wednesday October 19 2022

Time: 16:00 (CET)

Speaker: Prof. Sebastian Cioaba (University of Delaware)

Title: Spectral Moore theorems for graphs and hypergraphs

Abstract: The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander and Ramanujan graphs. In this paper, I will give an overview of recent work studying the maximum order of a regular graph (bipartite graph or hypergraph) of given valency whose second largest eigenvalue is at most a given value. This problem can be seen as a spectral Moore problem and has close connections to Alon-Boppana theorems for graphs and hypergraphs and with the usual Moore or degree-diameter problem

Slides: TBA

Video: https://drive.google.com/file/d/1O1o916Iu_XbVqIAR5dTAth3LOrLJFv9y/view?usp=sharing

Talk 18: On hamiltonian paths of the complete graph with prescribed edge-lengths - Prof. Anita Pasotti

Date: Wednesday November 16 2022

Time: 14:30 (CET)

Speaker: Prof. Anita Pasotti (University of Brescia, Italy)

Title: On hamiltonian paths of the complete graph with prescribed edge-lengths

Abstract: https://drive.google.com/file/d/1H07eEmsU9RsUwlVTb1uZTEajCA4F3lqu/view?usp=share_link

Slides: https://drive.google.com/file/d/1lltYz-fkvBxAV1XcN8uYENPXbX4L5QEc/view?usp=share_link

Video: https://drive.google.com/file/d/1GW1a0DjBuzI6wp3pceagI9KyY7kCZdSN/view?usp=share_link

Talk 19: Mixed graphs with small excess - Prof. Grahame Erskine

Date: Wednesday December 21 2022

Time: 16:00 CET (15:00 UK time)

Speaker: Prof. Grahame Erskine (The Open University, UK)

Title: Mixed graphs with small excess

Abstract: The degree-diameter problem for undirected graphs extends in a natural way to the cases of directed or mixed graphs; we insist that every vertex should be reachable from any other vertex by a path of speficied maximum length. However, the undirected degree-girth problem can be generalised in a number of ways. One natural extension to digraphs and mixed graphs is to insist that any pair of vertices be joined by no more than one path of some specified maximum length; this is called the geodecity problem.

The degree-geodecity problem is in a sense dual to the degree-diameter problem. They share the same Moore bound; in the case of the geodecity problem the Moore bound is a lower bound on the order of a graph with given degree and geodecity, and we are interested in minimising the excess of a graph above this bound. The problem has some features in common with the degree-girth problem for undirected graphs; chief among these is that it seems to be very hard to get good bounds on how small the excess of a directed or mixed graph can be for given degree and geodecity.

We will discuss some of what is known about this problem, with a particular emphasis on mixed graphs.

Slides: https://drive.google.com/file/d/1RkDrLfG3TfYqRY_kMesv9WPrEy4bcTOz/view?usp=share_link

Video: https://drive.google.com/file/d/1xumoar4OhI7l2oEZlDf4aFQB0rcnEn9g/view?usp=share_link

Talk 20: Approaching the Moore bound in the degree-diameter problem by Cayley graphs - Prof. Jana Šiagiová

Date: Wednesday January 18 2023

Time: 15:00 (CET)

Speaker: Prof. Jana Šiagiová (Slovak University of Technology in Bratislava)

Title: Approaching the Moore bound in the degree-diameter problem by Cayley graphs

Abstract: The well known degree-diameter problem for graphs is to determine the largest number n(d,k) of vertices in a graph of given maximum degree d and given diameter k. A trivial upper bound M(d,k) on n(d,k) is provided by the number of vertices of a rooted tree of the same maximum degree and of depth equal to the given diameter, and it is also well known that this upper bound is attained in a non-trivial way only for diameter 2 and degrees 3, 7, and possibly 57, and for no degree and diameter both greater than 2. Some years ago a natural question arose if examples of `large' graphs for given d and k can be constructed by means of Cayley graphs. We have answered this question in the affirmative for diameters k=2 and k=3 by proving that in each case there is an infinite set D(k) of degrees, and for each such degree d a Cayley graph with c(d,k) vertices, such that lim sup c(d,k)/M(d,k) is equal to 1 as d ranges over the set D(k). Thus, at least in this sense, for diameters 2 and 3 the Moore bound can be approached asymptotically. The aim of the talk is to present background on the methods and more details used in the proof of this asymptotic result, hoping to revive interest in trying to push it further.

Slides: []

Video: https://drive.google.com/file/d/1tQVeyG8oRfMfeJNCORLLvZaxlYB40r_x/view?usp=share_link

Talk 21: L(h,k)-colorings of some Moore graphs via some special structures - Prof. Julián Fresán

Date: Wednesday Frebruary 15, 2023

Time: 17:00 (CET)

Speaker: Prof. Julián Fresán (Universidad Autónoma Metropolitana-Cuajimalpa, México)

Title: L(h,k)-colorings of some Moore graphs via some special structures

Abstract: In this talk, I will present a particular structure of the non-adjacencies of the projective planes and two special structures of classical generalized quadrangles. We will use these structures to obtain bounds of a coloring problem, the L(h,k)-coloring problem, of their incidence graphs. This coloring is a vertex coloring with numbers in which the difference between any pair of vertices at distance one is at least h and any pair of vertices at distance two have coloring numbers that differ by at least k. If we start the coloring with zero, the goal of this problem is to find the L(h, k)-coloring with the smallest maximum number.

Slides: https://drive.google.com/file/d/10odT6l8zRF3DGaA899ljikoxEblQPW2d/view?usp=share_link

Video: https://drive.google.com/file/d/10lKFgZ3Nuvt1hqBRZisOMXmEeRO--Xni/view?usp=share_link

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)