A cycle cover of a 2-edge-connected graph embedded with large face-width on an orientable surface

2016 ◽  
Vol 212 (1) ◽  
pp. 219-235
Author(s):  
Dengju Ma ◽  
Han Ren
Keyword(s):  
Connected Graph ◽  
Orientable Surface ◽  
Cycle Cover ◽  
Face Width ◽  
Large Face

scholarly journals The Crossing Number of a Projective Graph is Quadratic in the Face–Width

10.37236/770 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
I. Gitler ◽  
P. Hliněný ◽  
J. Leaños ◽  
G. Salazar
Keyword(s):  
Approximation Algorithm ◽  
Projective Plane ◽  
Crossing Number ◽  
Orientable Surface ◽  
Constant Factor ◽  
Bounded Degree ◽  
Face Width ◽  
Projective Graph ◽  
The Face ◽  
Large Face

We show that for each integer $g\geq0$ there is a constant $c_g > 0$ such that every graph that embeds in the projective plane with sufficiently large face–width $r$ has crossing number at least $c_g r^2$ in the orientable surface $\Sigma_g$ of genus $g$. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective graphs with bounded degree.

scholarly journals Asymptotic Enumeration of Labelled Graphs by Genus

10.37236/500 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Edward A. Bender ◽  
Zhicheng Gao
Keyword(s):  
Orientable Surface ◽  
Asymptotic Formulas ◽  
Asymptotic Enumeration ◽  
Connected Graphs ◽  
Face Width ◽  
Surface Maps ◽  
Connected Surface ◽  
Labelled Graphs ◽  
Large Face

We obtain asymptotic formulas for the number of rooted 2-connected and 3-connected surface maps on an orientable surface of genus $g$ with respect to vertices and edges simultaneously. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled $k$-connected graphs of orientable genus $g$ for $k\le3$.

scholarly journals Asymptotic properties of rooted 3-connected maps on surfaces

1996 ◽  
Vol 60 (1) ◽  
pp. 31-41 ◽  
Author(s):  
Edward A. Bender ◽  
Zhicheng Gao ◽  
L. Bruce Richmond ◽  
Nicholas C. Wormald
Keyword(s):  
Asymptotic Properties ◽  
Face Width ◽  
Almost All ◽  
Large Face

AbstractIn this paper we obtain asymptotics for the number of rooted 3-connected maps on an arbitrary surface and use them to prove that almost all rooted 3-connected maps on any fixed surface have large edge-width and large face-width. It then follows from the result of Roberston and Vitray [10] that almost all rooted 3-connected maps on any fixed surface are minimum genus embeddings and their underlying graphs are uniquely embeddable on the surface.

Uniqueness and minimality of large face-width embeddings of graphs

COMBINATORICA ◽  
1995 ◽  
Vol 15 (4) ◽  
pp. 541-556 ◽  
Author(s):  
Bojan Mohar
Keyword(s):  
Face Width ◽  
Large Face

scholarly journals Counting unicellular maps on non-orientable surfaces

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Olivier Bernardi ◽  
Guillaume Chapuy
Keyword(s):  
Connected Graph ◽  
Topological Type ◽  
Orientable Surface ◽  
Recurrence Equation ◽  
Asymptotic Formulas ◽  
Topological Disk ◽  
International Audience ◽  
Fixed Topology ◽  
Orientable Surfaces

International audience A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we give a bijective operation that relates unicellular maps on a non-orientable surface to unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable precubic (all vertices of degree 1 or 3) unicellular maps of fixed topology. We also determine asymptotic formulas for the number of all unicellular maps of fixed topology, when the number of edges goes to infinity. Our strategy is inspired by recent results obtained for the orientable case [Chapuy, PTRF 2010], but significant novelties are introduced: in particular we construct an involution which, in some sense, ``averages'' the effects of non-orientability. \par Une carte unicellulaire est le plongement d'un graphe connexe dans une surface, tel que le complémentaire du graphe est un disque topologique. On décrit une opération bijective qui relie les cartes unicellulaires sur une surface non-orientable aux cartes unicellulaires de type topologique inférieur, avec des sommets marqués. On en déduit une récurrence qui conduit à de (nouvelles) formules closes d'énumération pour les cartes unicellulaires précubiques (sommets de degré 1 ou 3) de topologie fixée. On obtient aussi des formules asymptotiques pour le nombre total de cartes unicellulaires de topologie fixée, quand le nombre d'arêtes tend vers l'infini. Notre stratégie est motivée par de récents résultats dans le cas orientable [Chapuy, PTRF, 2010], mais d'importantes nouveautés sont introduites: en particulier, on construit une involution qui, en un certain sens, "moyenne'' les effets de la non-orientabilité.

Combinatorial Oriented Maps

1979 ◽  
Vol 31 (5) ◽  
pp. 986-1004 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Connected Graph ◽  
Orientable Surface ◽  
The Other ◽  
Cyclic Order ◽  
Positive Sense

An orientable map is often presented as a realization of a finite connected graphGin an orientable surface so that the complementary domains ofG, the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while.On each edge ofGwe can recognize two opposite directed edges, or “darts”. Letθbe the permutation of the dart-setSthat interchanges each dart with its opposite. The darts radiating from a vertexvoccur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutationPofS. The choice ofPrather thanP–l, which corresponds to the other sense of rotation, makes the map “oriented”.

Effects of misalignment on load distribution in large face-width helical gears

2003 ◽  
Vol 217 (2) ◽  
pp. 93-98 ◽  
Author(s):  
J Haigh ◽  
J N Fawcett
Keyword(s):  
Load Distribution ◽  
Experimental Results ◽  
Helical Gears ◽  
Face Width ◽  
Stress Levels ◽  
The Uk ◽  
The University ◽  
Large Face

Experimental results, obtained from the UK 8 MW facility for gear research at the University of Newcastle, are presented, which show how the load distribution across a large face-width gear varies with misalignment. Tests were carried out over a range of torque conditions using pinion flanks which were crowned or which had lead modification. The crowned flanks were shown to be more tolerant of misalignment but, at low misalignments, gave slightly higher stress levels than those with lead modification.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals Privacy Preserving Distributed Summation in a Connected Graph

2020 ◽  
Vol 53 (2) ◽  
pp. 3445-3450
Author(s):  
Katrine Tjell ◽  
Rafael Wisniewski
Keyword(s):  
Connected Graph ◽  
Privacy Preserving
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