It follows from the classical theorem by Lebesgue (1940) on the structure of minor faces in 3-polytopes that every plane triangulation with minimum degree at least 4 has a 3-face for which the set of degrees of its vertices is majorized by one of the following sequences: ( 4 , 4 , ∞ ), ( 4 , 5 , 19 ), ( 4 , 6 , 11 ), ( 4 , 7 , 9 ), ( 5 , 5 , 9 ), ( 5 , 6 , 7 ). In 1999, Jendrol' gave the following description of faces: ( 4 , 4 , ∞ ), ( 4 , 5 , 13 ), ( 4 , 6 , 17 ), ( 4 , 7 , 8 ), ( 5 , 5 , 7 ), ( 5 , 6 , 6 ). Also, Jendrol' (1999) conjectured that there is a face of one of the types: ( 4 , 4 , ∞ ), ( 4 , 5 , 10 ), ( 4 , 6 , 15 ), ( 4 , 7 , 7 ), ( 5 , 5 , 7 ), ( 5 , 6 , 6 ). In 2002, Lebesgue's description was strengthened by Borodin to ( 4 , 4 , ∞ ), ( 4 , 5 , 17 ), ( 4 , 6 , 11 ), ( 4 , 7 , 8 ), ( 5 , 5 , 8 ), ( 5 , 6 , 6 ). In 2014, we obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': ( 4 , 4 , ∞ ), ( 4 , 5 , 11 ), ( 4 , 6 , 10 ), ( 4 , 7 , 7 ), ( 5 , 5 , 7 ), ( 5 , 6 , 6 ). The purpose of this paper is to give another tight description of faces in plane triangulations with minimum degree at least 4: ( 4 , 4 , ∞ ), ( 4 , 6 , 10 ), ( 4 , 7 , 7 ), ( 5 , 5 , 8 ), ( 5 , 6 , 7 ).

The weight w(e) of an edge e in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge e = uv is an (i, j)-edge if d(u) <= i and d(v) <= j. In 1940, Lebesgue proved that every NPM has a (3,11)-edge, or (4, 7)-edge, or (5,6)-edge, where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-polytope has an edge e with w(e) <= 13, which bound is sharp. Borodin (1987), answering Erdos' question, proved that every NPM has such an edge. Moreover, Borodin (1991) refined this by proving that there is either a (3, 10)-edge, or (4, 7)-edge, or (5, 6)-edge. Given an NPM, we observe some upper bounds on the minimum weight of all its edges, denoted by w, of those incident with a 3-face, w*, and those incident with two 3-faces, w**. In particular, Borodin (1996) proved that if w** = infinity, that is if an NPM has no edges incident with two 3-faces, then either w* <= 9 or w <= 8, where both bounds are sharp. The purpose of our note is to refine this result by proving that in fact w** = infinity implies either a (3, 6)- or (4, 4)-edge incident with a 3-face, or a (3, 5)-edge, which description is tight.

The degree d(x) of a vertex or face x in a graph G is the number of incident edges. A face f = v(1) center dot center dot center dot v(d(f)) in a graph G on the plane or other orientable surface is of type (k(1), k(2), ...) if d(vi) <= k(i) for each i. By delta we denote the minimum vertex-degree of G.It follows from the classical theorem by Lebesgue (1940) that every plane triangulation with delta >= 4 has a 3-face of types (4, 4, infinity), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), or (5, 6, 7). In 1999, Jendrol' gave a similar description: "(4, 4, infinity), (4, 5,13), (4, 6,17), (4, 7, 8), (5, 5, 7), (5, 6, 6)" and conjectured that "(4, 4, infinity), (4, 5,10), (4, 6,15), (4, 7, 7), (5, 5, 7), (5, 6, 6)" holds. In 2002, Lebesgue's description was strengthened by Borodin to "(4, 4, infinity), (4, 5, 17), (4, 6, 11), (4, 7, 8), (5, 5, 8), (5, 6, 6)". In 2014, we obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': "(4, 4, infinity), (4, 5,11), (4, 6,10), (4, 7, 7), (5, 5, 7), (5, 6, 6)", and recently proved another tight description of faces in plane triangulations with delta >= 4: "(4, 4, infinity), (4, 6,10), (4, 7, 7), (5, 5, 8), (5, 6, 7)".It follows from Lebesgue's theorem of 1940 that every plane 3-connected quadrangulation has a face of one of the types (3, 3, 3, infinity), (3, 3, 4, 11), (3, 3, 5, 7), (3, 4, 4, 5).Recently, we improved this description to "(3, 3, 3, infinity), (3, 3, 4, 9), (3, 3, 5, 6), (3, 4, 4, 5)", where all parameters except possibly 9 are best possible and 9 cannot go down below 8.In 1995, Avgustinovich and Borodin proved the following tight description of the faces of torus quadrangulations with delta >= 3: "(3, 3, 3, infinity), (3, 3, 4,10), (3,3,5,7), (3,3,6,6), (3,4,4,6), (4, 4, 4, 4)".Recently, we proved that every triangulation with delta >= 4 of the torus has a face of one of the types (4, 4, infinity), (4, 6,12), (4, 8, 8), (5, 5, 8), (5, 6, 7), or (6, 6, 6), which description is tight.The purpose of this paper is to prove that every graph with delta >= 4 that admits a closed 2-cell embedding on the torus has a face of one of the types (4, 4, 4, 4), (4, 4,infinity), (4, 5,16), (4, 6,12), (4, 8, 8), (5, 5, 8), (5, 6, 7), or (6, 6, 6), where all parameters are best possible.

The degree d ( x ) of a vertex or face x in a graph G is the number of incident edges. A face f = v 1 … v d ( f ) in a graph G on the plane or other orientable surface is of type ( k 1 , k 2 , … ), where k 1 ≤ k 2 ≤ …, if d ( v i ) ≤ k i for each i . By δ we denote the minimum vertex-degree of G . In 1989, Borodin confirmed Kotzig's conjecture of 1963 that every plane graph with minimum degree δ equal to 5 has a ( 5 , 5 , 7 )-face or a ( 5 , 6 , 6 )-face, where all parameters are tight. Recently, we proved that every torus triangulation with δ ≥ 5 has a face of one of the types ( 5 , 5 , 8 ), ( 5 , 6 , 7 ), or ( 6 , 6 , 6 ), which is tight. It follows from the classical theorem by Lebesgue (1940) that every plane triangulation with δ ≥ 4 has a 3-face of types ( 4 , 4 , ∞ ), ( 4 , 5 , 19 ), ( 4 , 6 , 11 ), ( 4 , 7 , 9 ), ( 5 , 5 , 9 ), or ( 5 , 6 , 7 ). In 1999, Jendrol' gave a similar description: “( 4 , 4 , ∞ ), ( 4 , 5 , 13 ), ( 4 , 6 , 17 ), ( 4 , 7 , 8 ), ( 5 , 5 , 7 ), ( 5 , 6 , 6 )” and conjectured that “( 4 , 4 , ∞ ), ( 4 , 5 , 10 ), ( 4 , 6 , 15 ), ( 4 , 7 , 7 ), ( 5 , 5 , 7 ), ( 5 , 6 , 6 )” holds. In 2002, Lebesgue's description was strengthened by Borodin to “( 4 , 4 , ∞ ), ( 4 , 5 , 17 ), ( 4 , 6 , 11 ), ( 4 , 7 , 8 ), ( 5 , 5 , 8 ), ( 5 , 6 , 6 )”. In 2014, Borodin and Ivanova obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': “( 4 , 4 , ∞ ), ( 4 , 5 , 11 ), ( 4 , 6 , 10 ), ( 4 , 7 , 7 ), ( 5 , 5 , 7 ), ( 5 , 6 , 6 )”, and recently proved another tight description of faces in plane triangulations with δ ≥ 4: “( 4 , 4 , ∞ ), ( 4 , 6 , 10 ), ( 4 , 7 , 7 ), ( 5 , 5 , 8 ), ( 5 , 6 , 7 )”. It follows from Lebesgue's theorem of 1940 that every plane quadrangulation with δ ≥ 3 has a face of one of the types ( 3 , 3 , 3 , ∞ ), ( 3 , 3 , 4 , 11 ), ( 3 , 3 , 5 , 7 ), ( 3 , 4 , 4 , 5 ). Recently, Borodin and Ivanova improved this description to “( 3 , 3 , 3 , ∞ ), ( 3 , 3 , 4 , 9 ), ( 3 , 3 , 5 , 6 ), ( 3 , 4 , 4 , 5 )”, where all parameters except possibly 9 are best possible and 9 cannot go down below 8. In 1995, Avgustinovich and Borodin proved the following tight description of the faces of torus quadrangulations with δ ≥ 3: “( 3 , 3 , 3 , ∞ ), ( 3 , 3 , 4 , 10 ), ( 3 , 3 , 5 , 7 ), ( 3 , 3 , 6 , 6 ), ( 3 , 4 , 4 , 6 ), ( 4 , 4 , 4 , 4 )”, which also holds for each higher surface provided that its quadrangulation is large enough. Recently, Borodin and Ivanova proved that every triangulation with δ ≥ 4 of the torus has a face of one of the types ( 4 , 4 , ∞ ), ( 4 , 6 , 12 ), ( 4 , 8 , 8 ), ( 5 , 5 , 8 ), ( 5 , 6 , 7 ), or ( 6 , 6 , 6 ), which description is tight. The purpose of this paper is to prove that every triangulation with δ ≥ 3 on the torus has a face of one of the types ( 3 , 6 , 24 ), ( 3 , 8 , 16 ), ( 3 , 12 , 12 ), ( 4 , 4 , ∞ ), ( 4 , 6 , 12 ), ( 4 , 8 , 8 ), ( 5 , 5 , 8 ), ( 5 , 6 , 7 ), or ( 6 , 6 , 6 ), where all parameters are best possible.

Let g ( k , t ) be the minimum integer such that every plane graph with girth g at least g ( k , t ), minimum degree δ = 2 and no ( k + 1 )-paths consisting of vertices of degree 2, where k ≥ 1, has a 3-vertex with at least t neighbors of degree 2, where 1 ≤ t ≤ 3. In 2015, Jendrol' and Maceková proved g ( 1 , 1 ) ≤ 7. Later on, Hudák et al. established g ( 1 , 3 ) = 10, Jendrol', Maceková, Montassier, and Soták proved g ( 1 , 1 ) ≥ 7, g ( 1 , 2 ) = 8 and g ( 2 , 2 ) ≥ 11, and we recently proved that g ( 2 , 2 ) = 11 and g ( 2 , 3 ) = 14. Thus g ( k , t ) is already known for k = 1 and all t . In this paper, we prove that g ( k , 1 ) = 3 k + 4, g ( k , 2 ) = 3 k + 5, and g ( k , 3 ) = 3 k + 8 whenever k ≥ 2.

Lebesgue (1940) proved that every plane graph with minimum degree delta at least 3 and girth g (the length of a shortest cycle) at least 5 has a path on three vertices (3-path) of degree 3 each. A description of 3-paths is tight if none of its parameter can be strengthened, and no triplet dropped. Borodin et al. (2013) gave a tight description of 3-paths in plane graphs with delta >= 3 and g >= 3, and another tight description was given by Borodin, Ivanova and Kostochka in 2017. In 2015, we gave seven tight descriptions of 3-paths when delta >= 3 and g >= 4. Furthermore, we proved that this set of tight descriptions is complete, which was a result of a new type in the structural theory of plane graphs. Also, we characterized (2018) all one-term tight descriptions if delta >= 3 and g >= 3. The problem of producing all tight descriptions for g >= 3 remains widely open even for delta >= 3. Eleven tight descriptions of 3-paths were obtained for plane graphs with delta = 2 and g >= 4 by Jendrol', Macekova, Montassier, and Sotak, four of which are descriptions for g >= 9. In 2018, Aksenov, Borodin and Ivanova proved nine new tight descriptions of 3-paths for delta = 2 and g >= 9 and showed that no other tight descriptions exist. Recently, we resolved the case g >= 8. The purpose of this paper is to give a complete list of 15 tight descriptions of 3-paths in the plane graphs with delta = 2 and g >= 7. (C) 2021 Elsevier B.V. All rights reserved.

The degree d of a vertex or face in a graph G is the number of incident edges. A face f = v(1)...v(d) in a plane or torus graph G is of type (k1, k2,.., kd) if d(vz) < k, for each i. By (5 we denote the minimum vertex-degree of G. In 1989, Borodin confirmed Kotzig's conjecture of 1963 that every plane graph with minimum degree (5 equal to 5 has a (5, 5, 7) -face or a (5, 6, 6) -face, where all parameters are tight. It follows from the classical theorem of Lebesgue (1940) that every plane quadrangulation with (5 > 3 has a face of one of the types (3, 3, 3, infinity), (3, 3, 4, 11), (3, 3, 5, 7), (3, 4, 4, 5). Recently, we improved this description to the following one: "(3, 3, 3, co), (3, 3, 4, 9), (3,3, 5, 6), (3, 4, 4, 5)", where all parameters except possibly 9 are best possible and 9 cannot go down below 8. In 1995, Avgustinovich and Borodin proved that every torus quadrangulation with (5 > 3 has a face of one of the following types: (3, 3, 3, co), (3, 3, 4, 10), (3, 3, 5, 7), (3, 3, 6, 6), (3, 4, 4, 6), (4, 4, 4, 4), where all parameters are best possible. The purpose of our note is to prove that every torus triangulation with (5 > 5 has a face of one of the types (5, 5, 8), (5, 6, 7), or (6, 6, 6), where all parameters are best possible.

A 3-path uvw is an (i, j, k)-path if d(u) <= i , d(v) <= j, and d(w) <= k, where d(x) is the degree of a vertex x. It is well-known that each 3-polytope has a vertex of degree at most 5, called minor. A description of 3-paths in a 3-polytope is minor or major if the central item of each its triplet is at most 5 or at least 6, respectively. Back in 1922, Franklin proved that each 3-polytope with minimum degree 5 has a (6, 5, 6)-path, which description is tight. Recently, Borodin and Ivanova extended Franklin's theorem by producing all the ten tight minor descriptions of 3-paths in the class P-4 of 3-polytopes with minimum degree at least 4. In 2016, Borodin and Ivanova proved that each polytope with minimum degree 5 has a (5, 6, 6)-path, and there exists no tight description of 3-paths in this class of 3-polytopes other than {(6, 5, 6)} and {(5, 6, 6)}. The purpose of this paper is to prove that there exist precisely the following four major tight descriptions of 3-paths in P-4 : {(4, 9, 4), (4, 7, 5), (5,6,6)}, {(4,9,4), (5,7,6)}, (4,9,5), (5, 6, 6)}, and {(5,9,6)}.

Lebesgue (1940) proved that every plane graph with minimum degree δ at least 3 and girth g (the length of a shortest cycle) at least 5 has a path on three vertices (3-path) of degree 3 each. A description is tight if no its parameter can be strengthened, and no triplet dropped. Borodin et al. (2013) gave a tight description of 3-paths in plane graphs with δ ≥ 3 and g ≥ 3, and another tight description was given by Borodin, Ivanova and Kostochka in 2017. In 2015, we gave seven tight descriptions of 3-paths when δ ≥ 3 and g ≥ 4. Furthermore, we proved that this set of tight descriptions is complete, which was a result of a new type in the structural theory of plane graphs. Also, we characterized (2018) all one-term tight descriptions if δ ≥ 3 and g ≥ 3. The problem of producing all tight descriptions for g ≥ 3 remains widely open even for δ ≥ 3. Recently, eleven tight descriptions of 3-paths were obtained for plane graphs with δ = 2 and g ≥ 4 by Jendrol’, Maceková, Montassier, and Soták, four of which descriptions are for g ≥ 9. In 2018, Aksenov, Borodin and Ivanova proved nine new tight descriptions of 3-paths for δ = 2 and g ≥ 9 and showed that no other tight descriptions exist. The purpose of this note is to give a complete list of tight descriptions of 3-paths in the plane graphs with δ = 2 and g ≥ 8.

We consider plane graphs with large enough girth g, minimum degree delta at least 2 and no (k + 1)-paths consisting of vertices of degree 2, where k >= 1. In 2016, Hudak, Macekova, Madaras, and Siroczki studied the case k = 1, which means that no two 2-vertices are adjacent, and proved, in particular, that there is a 3-vertex whose all three neighbors have degree 2 (called a soft 3-star), provided that g >= 10, which bound on g is sharp. For the first open case k = 2 it was known that a soft 3-star exists if g >= 14 but may not exist if g <= 12. In this paper, we settle the case k = 2 by presenting a construction with g = 13 and no soft 3-star. For all k >= 3, we prove that soft 3-stars exist if g >= 4k + 6 but, as follows from our construction, possibly not exist if g <= 3k + 7. We conjecture that in fact soft 3-stars exist whenever g >= 3k + 8.

The long periods/phases (with a duration of more than 10-15 years) of increased and decreased water flow of rivers in the Volga river basin were identified. The annual and seasonal water flow (over snow-melt flood period, as well as summer–autumn and winter low-water seasons) of six representative rivers over observation periods more than a century in duration, starting from the 1870s–1890s up to 2016, was used. In addition to this, periods with the average runoff close to its normal value were also observed. The boundaries of contrast phases were determined using normalized cumulative deviation curves in combination with Student’s test of the statistical homogeneity of the data series. The duration of the phases varies from 10 to 96 years. The phases of lower runoff were generally longer than those of higher runoff (this is especially typical of the winter and summer–autumn low-water season). The identified contrast phases show a statistically significant difference between the annual and seasonal runoff. The analysis of the data series of alteration of phases with increased and decreased water flow in hydrological seasons of the year allowed the authors to identify three major types of their long-term dynamics within the Volga river basin.

Back in 1922, Franklin proved that every 3-polytope with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions. It is well-known that each 3-polytope has a vertex of degree at most 5, called minor vertex. A 3-path uvw is an (i, j, k)-path if d(u) <= i, d(v) <= j, and d(w) <= k, where d(x) is the degree of a vertex x. A 3-path is minor 3-path if its central vertex is minor. The purpose of this note is to extend Franklin' Theorem to the 3-polytopes with minimum degree at least 4 by proving that there exist precisely the following ten tight descriptions of minor 3-paths: {(6, 5, 6), (4, 4, 9), (6, 4, 8), (7, 4, 7)}, {(6, 5, 6), (4, 4, 9), (7, 4, 8)}, {(6, 5, 6), (6, 4, 9), (7, 4, 7)}, {(6, 5, 6), (7, 4, 9)}, {(6, 5, 8), (4, 4, 9), (7, 4, 7)}, {(6, 5, 9), (7, 4, 7)}, {(7, 5, 7), (4, 4, 9), (6, 4, 8)}, {(7, 5, 7), (6, 4, 9)}, {(7, 5, 8), (4, 4, 9)}, and {(7, 5, 9)}.

The degree of a vertex or face in a 3-polytope is the number of incident edges. A k-face is one of degree k, a k−-face has degree at most k. The height of a face is the maximum degree of its incident vertices; and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; and so we assume the absence of pyramidal faces in what follows. In 1940, Lebesgue proved that each quadrangulated 3-polytope has a face f with h(f) ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20, which bound is sharp. Later, Borodin proved that h ≤ 20 for all triangulated 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily polytopes that h ≤ 23. Recently, we obtained the sharp bounds h ≤ 10 for triangle-free polytopes and h ≤ 20 for arbitrary polytopes. Later, Borodin, Bykov, and Ivanova refined the latter result by proving that any polytope has a 10−-face of height at most 20, where 10 and 20 are sharp. Also, we proved that any polytope has a 5−-face of height at most 30, where 30 is sharp and improves the upper bound of 39 obtained by Horňák and Jendrol’ (1996). In this paper we prove that every polytope has a 6−-face of height at most 22, where 6 and 22 are best possible. Since there is a construction in which every face of degree from 6 to 9 has height 22, we now know everything concerning the maximum heights of restricted-degree faces in 3-polytopes.

Lebesgue (1940) proved that every plane graph with minimum degree delta at least 3 and girth g (the length of a shortest cycle) at least 5 has a path on three vertices (3-path) of degree 3 each. A description is tight if no its parameter can be strengthened, and no triplet dropped. Borodin et al. (2013) gave a tight description of 3-paths in plane graphs with delta >= 3 and g >= 3, and another tight description was given by Borodin, Ivanova and Kostochka in 2017. In 2015, we gave seven tight descriptions of 3-paths when delta >= 3 and g >= 4. Furthermore, we proved that this set of tight descriptions is complete, which was a result of a new type in the structural theory of plane graphs. Also, we characterized (2018) all one-term tight descriptions if delta >= 3 and g >= 3. The problem of producing all tight descriptions for g >= 3 remains widely open even for delta >= 3. Recently, eleven tight descriptions of 3-paths were obtained for plane graphs with delta = 2 and g > 4 by Jendrol', Macekova, Montassier, and Sotak, four of which descriptions are for g >= 9. In 2018, Aksenov, Borodin and Ivanova proved ten new tight descriptions of 3-paths for delta = 2 and g >= 9 and showed that no other tight descriptions exist. In this paper we give a complete list of tight descriptions of 3-paths centered at a 2-vertex in the plane graphs with delta = 2 and g >= 6.

In 1940, Lebesgue proved that every 3-polytope with minimum degree 5 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: ( 6 , 6 , 7 , 7 , 7 ) , ( 6 , 6 , 6 , 7 , 9 ) , ( 6 , 6 , 6 , 6 , 11 ) , ( 5 , 6 , 7 , 7 , 8 ) , ( 5 , 6 , 6 , 7 , 12 ) , ( 5 , 6 , 6 , 8 , 10 ) , ( 5 , 6 , 6 , 6 , 17 ) , ( 5 , 5 , 7 , 7 , 13 ) , ( 5 , 5 , 7 , 8 , 10 ) , ( 5 , 5 , 6 , 7 , 27 ) , ( 5 , 5 , 6 , 6 , ∞ ) , ( 5 , 5 , 6 , 8 , 15 ) , ( 5 , 5 , 6 , 9 , 11 ) , ( 5 , 5 , 5 , 7 , 41 ) , ( 5 , 5 , 5 , 8 , 23 ) , ( 5 , 5 , 5 , 9 , 17 ) , ( 5 , 5 , 5 , 10 , 14 ) , ( 5 , 5 , 5 , 11 , 13 ) . Recently, we proved that forbidding vertices of degree from 7 to 11 results in a tight description ( 5 , 5 , 6 , 6 , ∞ ), ( 5 , 6 , 6 , 6 , 15 ), ( 6 , 6 , 6 , 6 , 6 ). The purpose of this note is to prove that every 3-polytope with minimum degree 5 and no vertices of degrees 6, 7, and 8 has a 5-vertex whose neighborhood is majorized by one of the sequences ( 5 , 5 , 5 , 5 , ∞ ) and ( 5 , 5 , 5 , 10 , 12 ), which is tight and improves a corresponding description ( 5 , 5 , 5 , 5 , ∞ ), ( 5 , 5 , 9 , 5 , 17 ), ( 5 , 5 , 10 , 5 , 14 ), ( 5 , 5 , 11 , 5 , 13 ) that follows from the Lebesgue Theorem.

It is trivial that every 3-polytope has a face of degree at most 5, called minor. Back in 1940, Lebesgue gave an approximate description of minor faces in 3-polytopes, depending on 17 main parameters. In 2002, Borodin improved Lebesgue's description on six parameters without worsening the others and suggested to find a tight description of minor faces. So far, such a tight description has been obtained only for several restricted classes of 3-polytopes: those with minimum degree 5 (Borodin, 1989), without vertices of degree 3 (Borodin and Ivanova, 2013), for plane triangulations (Borodin et al. 2014), and without vertices of degree from 4 to 7 (Borodin et al. 2017). In this paper, we consider 3-polytopes without vertices of degree from 5 to 7. A face is of type (k(1), k(2), ...) if the set of degrees of its incident vertices is majorized by the vector (k(1), k(2), ...). It follows from results by Honiak and Jendrol' (1996) that every such polytope has a face of one of the types (4, 4, infinity), (3, 8, 15), (3, 9, 14), (3, 10, 13), (3, 3, 3, infinity), (3, 3, 4, 11), (3, 4, 4, 4), and (3, 3, 3, 3, 4), which improves four parameters in what can be obtained from Lebesgue's description. We prove a tight description "(4, 4, infinity), (3, 8, 14), (3, 9, 13), (3, 10, 12), (3, 3, 3, infinity), (3, 3, 4, 11), (3, 4, 4, 4), (3, 3, 3, 3, 4)" and, in particular, give constructions (unexpected for us) showing that 13 and 11 here are best possible (constructions confirming the sharpness of the other parameters were known before). (C) 2019 Elsevier B.V. All rights reserved.

It is trivial that every 3-polytope has a face of degree at most 5, called minor. The height h(f) of a face f is the maximum degree of the vertices incident with f It follows from the partial double n-pyramids that h(f) can be arbitrarily large for each f if a 3-polytope is allowed to have faces of types (4, 4, infinity) or (3, 3, 3, infinity). In 1996, M. Hornak and S. Jendrol' proved that every 3-polytope without faces of types (4, 4, infinity) and(3, 3, 3, infinity) has a minor face of height at most 39 and constructed such a 3-polytope satisfying h(f) >= 30 for all minor faces f. The purpose of this paper is to prove that every 3-polytope without faces of types (4, 4, infinity) and(3, 3, 3, infinity) has a minor face of height at most 30, which bound is tight due to the Hornak-Jendrol' construction. (C) 2018 Elsevier B.V. All rights reserved.

Abstract In 1940, Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P5 of 3-polytopes with minimum degree 5. Given a 3-polytope P, by h5(P) we denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P. Recently, Borodin, Ivanova and Jensen showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to two 5-vertices and two more vertices of degree at most 6, called a (5, 5, 6, 6, ∞)-vertex, then h5(P) can be arbitrarily large. Therefore, we consider the subclass P*5 of 3-polytopes in P5 that avoid (5, 5, 6, 6, ∞)-vertices. For each P*in P*5 without vertices of degree from 7 to 9, it follows from Lebesgue’s Theorem that h5(P*) ≤ 17. Recently, this bound was lowered by Borodin, Ivanova, and Kazak to the sharp bound h5(P*) ≤ 15 assuming the absence of vertices of degree from 7 to 11 in P*. In this note, we extend the bound h5(P*) ≤ 15 to all P*s without vertices of degree from 7 to 9.

In the article, it is considered a modification of an integral model of unsteady turbulent jet with a presence of pressure force. Stationary solutions of the presented model is compared with well-known analytical results of classical models. It is shown that the inclusion of pressure forces changes dynamic parameters of a jet by about 15%. An analytical solution of a steady forced buoyant jet that corresponds to a volcanic outburst is deduced. An analytical solution for the spontaneous jet of convective surface layer is presented. The simplest model of an ensemble of the buoyant jets of convective surface layer is built. A hydrodynamic formation mechanism of vertical profiles of the turbulent diffusivity and the turbulent statistical moments of the atmospheric surface layer related to the ascent of the jets’ system, is formulated.

Lebesgue (1940) proved that every plane graph with minimum degree delta at least 3 and girth g at least 5 has a path on three vertices (3-path) of degree 3 each. A description is tight if no its parameter can be strengthened, and no triplet dropped.Borodin et al. (2013) gave a tight description of 3-paths in plane graphs with delta >= 3 and g >= 3, and another tight description was given by Borodin, Ivanova and Kostochka in 2017.Borodin and Ivanova (2015) gave seven tight descriptions of 3-paths when delta >= 3 and g >= 4. Furthermore, they proved that this set of tight descriptions is complete, which was a result of a new type in the structural theory of plane graphs. Also, they characterized (2018) all oneterm tight descriptions if delta >= 3 and g >= 3. The problem of producing all tight descriptions for g >= 3 remains widely open even for delta >= 3.Recently, several tight descriptions of 3-paths were obtained for plane graphs with delta = 2 and g >= 4 by Jendrol', Macekova, Montassier, and Sotak, four of which descriptions are for g >= 9.In this paper, we prove ten new tight descriptions of 3-paths for delta = 2 and g >= 9 and show that no other tight descriptions exist.