# Raziskovalni matematični seminar - Arhiv

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**Datum in ura**/ Date and time: 26.4.19

**Predavalnica**/ Location: FAMNIT-MP6

**Predavatelj**/ Lecturer: Riste Škrekovski (UL FMF, UP FAMNIT, IMFM, FIŠ)

**Naslov**/ Title: Some results and problems on unique-maximum colorings of plane graphs

**Vsebina**/ Abstract:

A unique-maximum coloring of a plane graph G is a proper vertex coloring by natural numbers such that each face \alpha of G satisfies the property: the maximal color that appears on \alpha, appears precisely on one vertex of \alpha (or shortly, the maximal color on every face is unique on that face). Fabrici and G\"{o}ring proved that six colors are enough for any plane graph and conjectured that four colors suffice. Thus, this conjecture is a strengthening of the Four Color Theorem. Wendland later decreased the upper bound from six to five.

We first show that the conjecture holds for various subclasses of planar graphs but then we disprove it for planar graphs in general. Thus, the facial unique-maximum chromatic number of the sphere is not four but five. In the second part of the talk, we will consider various new directions and open problems.

(Joint work with Vesna Andova, Bernard Lidick\'y, Borut Lužar, and Kacy Messerschmidt)

**Datum in ura**/ Date and time: 15.4.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Graziano Gentili (Universita degli studi Firenze, Italy)

**Naslov**/ Title: Regular functions of a quaternionic variable with applications

**Vsebina**/ Abstract:

The recent notion of quaternionic slice regularity has identified a new class of functions which can play the role of holomorphic (and meromorphic) functions in the quaternionic setting. The talk will present the main basic features of slice regular functions over the quaternions, as well as some of their most curious applications, including those concerning the study of orthogonal complex structures on subsets of the quaternionic space H.

**Datum in ura**/ Date and time: 8.4.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Berenice Martinez-Barona (Universitat Politecnica de Catalunya and University of Primorska)

**Naslov**/ Title: Identifying codes in digraphs

**Vsebina**/ Abstract:

A (1,\le \ell)-identifying code in a digraph D is a dominating subset C of vertices of D such that all distinct subsets of vertices of cardinality at most \ell have distinct closed in-neighbourhoods within C.

In this talk, we give an upper bound on \ell and some sufficient conditions for a digraph of minimum in-degree \delta^-\ge 1 to admit a (1,\le \ell)-identifying code for \ell= \delta^-, \delta^-+1. We give also a new method to obtain an upper bound on \ell for digraphs and graphs using spectral graph theory. As particular cases, we give a characterization of the j-in-regular digraphs admitting a (1,\le \ell)-identifying code for j \in \{1,2\} and \ell\in\{1,2,3\}. We also prove that every k-iterated line digraph of minimum in-degree at least 2 admits a (1,\le \ell)-identifying code with \ell \le 2, and it does not admit a (1,\le \ell)-identifying code for \ell\ge 3. Moreover, we give some bounds of the identifying number of a line digraph, that is, the minimum size of a (1,\le 1)-identifying code.