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Evgeniy Krasko

Publications and source records attributed to Evgeniy Krasko.

7 recordsLinked to original sources
arXiv2019

Enumeration of Unsensed Orientable Maps on Surfaces of a Given Genus

In this work for the first time we enumerate unlabelled maps on orientable genus $g$ surfaces with respect to all homeomorphisms, including both orientation-preserving and orientation-reversing. We show that in the latter case as an intermediate step one has to enumerate rooted maps of a special kind (quotient maps) on orientable and non-orientable surfaces possibly having a boundary and a certain number of branch points. In this work we develop a special technique for enumerating such maps.

math.CO↗
arXiv2018

Counting Unlabelled Chord Diagrams of Maximal Genus

Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces and a certain class of symmetric chord diagrams. This result extends the one of Cori and Marcus regarding maximal chord diagrams enumerated up to rotations.

math.CO↗
arXiv2017

Enumeration of $r$-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps

The work that consists of two parts is devoted to the problem of enumerating unrooted $r$-regular maps on the torus up to all its symmetries. We begin with enumerating near-$r$-regular rooted maps on the torus, projective plane and the Klein bottle. We also present the results of enumerating some special kinds of maps on the sphere: near-$r$-regular maps, maps with multiple leaves and maps with multiple root semi-edges. For $r=3$ and $r=4$ we obtain exact analytical formulas. For larger $r$ we derive recurrence relations. Then using these results we enumerate $r$-regular maps on the torus up to homeomorphisms that preserve its orientation --- so-called sensed maps. Using the concept of a quotient map on an orbifold we reduce this problem to enumeration of certain classes of rooted maps. For $r=3$ and $r=4$ we obtain closed-form expressions for the numbers of $r$-regular sensed maps by edges. All these results will be used in the second part of the work to enumerate $r$-regular maps on the torus up to all homeomorphisms --- so-called unsensed maps.

math.CO↗
arXiv2017

Enumeration of $r$-regular Maps on the Torus. Part II: Enumeration of Unsensed Maps

The second part of the paper is devoted to enumeration of $r$-regular toroidal maps up to all homeomorphisms of the torus (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to these homeomorphisms leads to maps on the Klein bottle, annulus and the Möbius band. Using $3$- and $4$-regular maps as an example we describe the technique of enumerating quotient maps on surfaces with a boundary. Obtained recurrence relations are used to enumerate unsensed $r$-regular maps on the torus for various $r$.

math.CO↗
arXiv2016

Enumeration of Chord Diagrams without Loops and Parallel Chords

We enumerate chord diagrams without loops and without both loops and parallel chords. We show that the former ones describe Hamiltonian paths in $n$-dimensional octahedrons. The latter ones are also known as shapes. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations.

math.CO↗