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Proper Orientations of Planar Bipartite Graphs
[article]

2016
*
arXiv
*
pre-print

As a consequence

arXiv:1609.06778v1
fatcat:fhouvu7k75gr7kn2o2mh6mwepm
*of*this result, we prove that 3-connected*planar*bipartite*graphs*have proper*orientation*number at most 6. ... The proper*orientation*number χ(G)*of*a*graph*G is the minimum*of*the maximum indegree, taken over all proper*orientations**of*G. ... We observe also that a*2*-*orientation**of*a bipartite*planar**graph*G can be found in time O(n*2*). ...##
###
Rectilinear planar layouts and bipolar orientations of planar graphs

1986
*
Discrete & Computational Geometry
*

The algorithm is based on the concept

doi:10.1007/bf02187706
fatcat:nwavk74fsjetbafomsdcydurnm
*of*a bipolar*orientation*. We discuss relationships among the bipolar*orientations**of*a*planar**graph*. I. ... We propose a linear-time algorithm for generating a*planar*layout*of*a*planar**graph*. Each vertex is represented by a horizontal line segment and each edge by a vertical line segment. ... Fig.*2*.*2*The layout*of*the*graph*inFig.*l*(a)implied by the selected bipolar*orientation*. The vertices have been given positive thickness for clarity. . ...##
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Colorings and girth of oriented planar graphs

1997
*
Discrete Mathematics
*

We investigate here homomorphisms from

doi:10.1016/s0012-365x(96)00198-7
fatcat:jue4vhi7tfeppdyyg7qe5hu6ra
*orientations**of*undirected*planar**graphs*to*graphs*(not necessarily*planar*) containing as few digons as possible. ... In particular, we prove that every*orientation**of*any large girth*planar**graph*is Scolorable and classify those digraphs on 3, 4 and 5 vertices which color all large girth*oriented**planar**graphs*. ... Proof*of*Theorem*l*(*1*) and (*2*). (*1*) Suppose that there exists a tournament T4 on 4 vertices such that every*oriented**planar**graph*G with sufficiently large girth has T, as a homomorphic image. ...##
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On the Number of Upward Planar Orientations of Maximal Planar Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

We consider the problem

doi:10.1007/978-3-642-35261-4_44
fatcat:yxunzi5uizbnhehk7sktc7mhy4
*of*determining the maximum and the minimum number*of*upward*planar**orientations*a maximal*planar**graph*can have. ... Moreover, there exist n-vertex maximal*planar**graphs*having as few as O(n ·*2*n ) upward*planar**orientations*and n-vertex maximal*planar**graphs*having Ω(2.599 n ) upward*planar**orientations*. ... Thus the total number*of*upward*planar**orientations**of*G satisfying Properties (*1*) and (*2*) is at most k−*1*j=p |S(j)| ≤ k−*1*j=p 4*l*−*1**2*(k+*1*)−j = 4*l*−*1**2*k−p+*1*k−p−*1*j=0*2*−j < 4*l**2*k−p . ...##
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On the number of upward planar orientations of maximal planar graphs

2014
*
Theoretical Computer Science
*

We consider the problem

doi:10.1016/j.tcs.2014.04.024
fatcat:gkwwxhkj4vgllne4uukg7u4vxy
*of*determining the maximum and the minimum number*of*upward*planar**orientations*a maximal*planar**graph*can have. ... Moreover, there exist n-vertex maximal*planar**graphs*having as few as O(n ·*2*n ) upward*planar**orientations*and n-vertex maximal*planar**graphs*having Ω(2.599 n ) upward*planar**orientations*. ... Thus the total number*of*upward*planar**orientations**of*G satisfying Properties (*1*) and (*2*) is at most k−*1*j=p |S(j)| ≤ k−*1*j=p 4*l*−*1**2*(k+*1*)−j = 4*l*−*1**2*k−p+*1*k−p−*1*j=0*2*−j < 4*l**2*k−p . ...##
###
Colorings of oriented planar graphs avoiding a monochromatic subgraph
[article]

2022
*
arXiv
*
pre-print

, every

arXiv:2102.07705v2
fatcat:dyilpan7kfa63am4w6ejg3caya
*oriented**planar**graph*D admits an F-free 3-coloring. ... We show in contrast, that - if F is an*orientation**of*a path*of*length at least*2*, then it is NP-hard to decide whether an acyclic and*planar*input digraph D admits an F-free*2*-coloring. - if F is an*orientation*... Introduction In this paper we are concerned with colorings*of*simple*planar*digraphs, i.e.,*orientations**of**planar**graphs*. We consider the following notion*of*coloring. Definition*1*. ...##
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Minimizing the Oriented Diameter of a Planar Graph

2009
*
Electronic Notes in Discrete Mathematics
*

We consider the problem

doi:10.1016/j.endm.2009.07.043
fatcat:7gt7miinencixarnot4pa6wa4a
*of*minimizing the diameter*of*an*orientation**of*a*planar**graph*. ... In contrast to this, for each constant*l*, we describe an algorithm that decides if a*planar**graph*G has an*orientation*with diameter at most*l*and runs in time O(c|V |), where c depends on*l*. ... (*l*+*1*) 2k+*2*h(*l*, k)*2*) from c(v*1*) and c(v*2*); • from c(r) we can determine in time O(k(*l*+*1*) k+*1*h(*l*, k)) whether the*graph*can be*oriented*as required. ...##
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On universal graphs for planar oriented graphs of a given girth

1998
*
Discrete Mathematics
*

We prove here that (

doi:10.1016/s0012-365x(97)00276-8
fatcat:7rmeiloc3jcsvj5egmtj535iqa
*1*) there exist no*planar*~4-universal*graphs*; (*2*) there exists a*planar*~16-universal*graph*on 6 vertices; (3) for any k, there exist no*planar*~k-universal*graphs**of*girth at least ... The*oriented*chromatic number o(H)*of*an*oriented**graph*H is defined to be the minimum order*of*an*oriented**graph*H' such that H has a homomorphism to H'. ... Denote by ~k the class*of**planar**oriented**graphs*with girth at least k. In particular, ~3 is the class*of*all*planar**oriented**graphs*. ...##
###
(E)-1-(2-Nitrobenzylidene)-2-phenylhydrazine

2010
*
Acta Crystallographica Section E
*

In the crystal, both molecules form inversion dimers linked by pairs

doi:10.1107/s1600536810025882
pmid:21588279
pmcid:PMC3007397
fatcat:4ck6yrjkl5b2nlzuall4vwpn6a
*of*N-HÁ Á ÁO hydrogen bonds, thereby generating R*2**2*(16) rings. ... For*graph*-set notation, see: Bernstein et al. (1995). ... The authors acknowledge the provision*of*funds for the purchase*of*the diffractometer and encouragement by Dr Muhammad Akram Chaudhary, Vice Chancellor, University*of*Sargodha, Pakistan. ...##
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(E)-3-(2-Nitrophenyl)-1-{1-phenylsulfonyl-2-[(phenylsulfonyl)methyl]-1H-indol-3-yl}prop-2-en-1-one

2011
*
Acta Crystallographica Section E
*

The central prop-

doi:10.1107/s1600536811051026
pmid:22259588
pmcid:PMC3254289
fatcat:xdy4753a7zhy7f5gfi734ox6ge
*2*-ene-*1*-one group is*oriented*at a dihedral angle*of*44.26 (11) with respect to the nitrophenyl ring and at 39.24 (8) with respect to the indole unit. ... In the crystal, molecules are linked into centrosymmetric dimers via pairs*of*C-HÁ Á ÁO hydrogen bonds with an R*2**2*(24)*graph*-set motif. ... The central prop-*2*-ene-*1*-one group is*oriented*at a dihedral angle*of*44.26 (11) with respect to the nitrophenyl ring and at 39.24 (8) with respect to the indole unit. ...##
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Oriented coloring of triangle-free planar graphs and 2-outerplanar graphs

2011
*
Electronic Notes in Discrete Mathematics
*

*Oriented*colouring

*of*

*2*-outerplanar

*graphs*, Inform. ... We also prove that every

*oriented*

*2*-outerplanar

*graph*has an

*oriented*chromatic number at most 40, that improves the previous known bound

*of*67 [Esperet,

*L*. and Ochem, P. ... In this paper, we first describe the construction

*of*the

*graph*T 40 in Section

*2*, an

*oriented*

*graph*on 40 vertices which has very useful properties for

*oriented*coloring

*of*

*planar*

*graphs*. ...

##
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Oriented Coloring of Triangle-Free Planar Graphs and 2-Outerplanar Graphs

2013
*
Graphs and Combinatorics
*

*Oriented*colouring

*of*

*2*-outerplanar

*graphs*, Inform. ... We also prove that every

*oriented*

*2*-outerplanar

*graph*has an

*oriented*chromatic number at most 40, that improves the previous known bound

*of*67 [Esperet,

*L*. and Ochem, P. ... In this paper, we first describe the construction

*of*the

*graph*T 40 in Section

*2*, an

*oriented*

*graph*on 40 vertices which has very useful properties for

*oriented*coloring

*of*

*planar*

*graphs*. ...

##
###
The 1-2 model
[article]

2015
*
arXiv
*
pre-print

The

arXiv:1507.04109v2
fatcat:n6mu5cl7fbbx3dqbvjyxcwjkhy
*1*-*2*model on the hexagonal lattice is a model*of*statistical mechanics in which each vertex is constrained to have degree either*1*or*2*. ... The current paper is a short review*of*rigorous results for the*1*-*2*model. ... Acknowledgements This work was supported in part by the Engineering and Physical Sciences Research Council under grant EP/103372X/*1*. ...##
###
2-Amino-4-nitrophenol–1-(2,4,6-trihydroxyphenyl)ethanone (1/1)

2012
*
Acta Crystallographica Section E
*

In the title compound, C 6 H 6 N

doi:10.1107/s1600536812017497
pmid:22590392
pmcid:PMC3344630
fatcat:hj3orfak2zh7tiautboiwc6wbm
*2*O 3 ÁC 8 H 8 O 4 , the*2*-amino-4nitrophenol (ANP) and*1*-(2,4,6-trihydroxyphenyl)ethanone (THA) molecules are both nearly*planar*, with r.m.s. deviations*of*0.0630 and ... The angle between the least-squares planes*of*THA and ANP is 48.99 (*2*) . In THA, an intramolecular O-HÁ Á ÁO hydrogen bond generates an S(6) ring motif. ... UV/Visible (nm): 226 (2,212 Å; ε= 19230*L*mol -*1*cm -*1*) and 288 (1,815 Å; ε= 15780*L*mol -*1*cm -*1*) π→π * transitions*of*benzene ring (E bands), 380 nm (0,345 Å; ε= 3000*L*mol -*1*cm -*1*) π→π * transition ...##
###
1-(2,4-Dinitrophenyl)-2-(1,2,3,4-tetrahydronaphthalen-1-ylidene)hydrazine

2010
*
Acta Crystallographica Section E
*

The authors acknowledge the provision

doi:10.1107/s1600536810028680
pmid:21588393
pmcid:PMC3007257
fatcat:6lap373sczckzpteyhjczpysli
*of*funds for the purchase*of*diffractometer and encouragement by Dr Muhammad Akram Chaudhary, Vice Chancellor, University*of*Sargodha, Pakistan. ... For*graph*-set notation, see: Bernstein et al. (1995) . Experimental Crystal data (i) Àx þ*1*; y À*1**2*; Àz þ*1**2*. ... In (I), the group A (C1-C7/C10)*of*3,4-dihydronaphthalen-*1*(2H)-ylidene moiety is*planar*with r. m. s. deviation*of*0.0051 Å. ...
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