If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types th... more If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types then the map is said to be a Semi-equivelar map. In this article we classify all semi-equivelar and vertex transitive maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$.
If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is s... more If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is said to be a semi-equivelar map. In this article, we classify all the types of semi-equivelar maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$. That is, we present {a complete map types of} semi-equivelar maps (if exist) on the surface of Euler char. $-1$. We know the complete list of semi-equivelar maps (upto isomorphism) for some types. Here, we also present a complete list of semi-equivelar maps for one type and for other types, similar steps can be followed.
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. The... more We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types {3^6}, {4^4}, {6^3}, {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4}. We know the classification of the maps of types {3^6}, {4^4}, {6^3} on the torus. In this article, we attempt to classify maps of types {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4} on the torus.
A triangulation of a connected closed surface is called weakly regular if the action of its autom... more A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinct n-vertex weakly regular triangulations of the torus for each n ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for each m ≥ 2. For 12 ≤ n ≤ 15, we have classified ...
If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is s... more If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is said to be a semi-equivelar map. In this article, we classify all the types of semi-equivelar maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$. That is, we present {a complete map types of} semi-equivelar maps (if exist) on the surface of Euler char. $-1$. We know the complete list of semi-equivelar maps (upto isomorphism) for some types. Here, we also present a complete list of semi-equivelar maps for one type and for other types, similar steps can be followed.
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. The... more We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$, $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$. We know the classification of the maps of types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$ on the torus. In this article, we attempt to classify maps of types $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$ on the torus.
If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types th... more If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types then the map is said to be a Semi-equivelar map. In this article we classify all semi-equivelar and vertex transitive maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$.
If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is s... more If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is said to be a semi-equivelar map. In this article, we classify all the types of semi-equivelar maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$. That is, we present {a complete map types of} semi-equivelar maps (if exist) on the surface of Euler char. $-1$. We know the complete list of semi-equivelar maps (upto isomorphism) for some types. Here, we also present a complete list of semi-equivelar maps for one type and for other types, similar steps can be followed.
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. The... more We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types {3^6}, {4^4}, {6^3}, {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4}. We know the classification of the maps of types {3^6}, {4^4}, {6^3} on the torus. In this article, we attempt to classify maps of types {3^3, 4^2}, {3^2, 4, 3, 4}, {3, 6, 3, 6}, {3^4, 6}, {4, 8^2}, {3, 12^2}, {4, 6, 12}, {3, 4, 6, 4} on the torus.
A triangulation of a connected closed surface is called weakly regular if the action of its autom... more A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinct n-vertex weakly regular triangulations of the torus for each n ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for each m ≥ 2. For 12 ≤ n ≤ 15, we have classified ...
If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is s... more If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is said to be a semi-equivelar map. In this article, we classify all the types of semi-equivelar maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$. That is, we present {a complete map types of} semi-equivelar maps (if exist) on the surface of Euler char. $-1$. We know the complete list of semi-equivelar maps (upto isomorphism) for some types. Here, we also present a complete list of semi-equivelar maps for one type and for other types, similar steps can be followed.
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. The... more We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$, $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$. We know the classification of the maps of types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$ on the torus. In this article, we attempt to classify maps of types $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$ on the torus.
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