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### Proper Orientations of Planar Bipartite Graphs [article]

Fiachra Knox, Sebastián González Hermosillo de la Maza, Bojan Mohar, Cláudia Linhares Sales
2016 arXiv   pre-print
As a consequence 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

Pierre Rosenstiehl, Robert E. Tarjan
1986 Discrete & Computational Geometry
The algorithm is based on the concept 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. .  ...

### Colorings and girth of oriented planar graphs

Jarik Nešetřil, André Raspaud, Eric Sopena
1997 Discrete Mathematics
We investigate here homomorphisms from 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.  ...

### On the Number of Upward Planar Orientations of Maximal Planar Graphs [chapter]

Fabrizio Frati, Joachim Gudmundsson, Emo Welzl
2012 Lecture Notes in Computer Science
We consider the problem 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 l1 2 (k+1)−j = 4 l1 2 k−p+1 k−p−1 j=0 2 −j < 4 l 2 k−p .  ...

### On the number of upward planar orientations of maximal planar graphs

Fabrizio Frati, Joachim Gudmundsson, Emo Welzl
2014 Theoretical Computer Science
We consider the problem 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 l1 2 (k+1)−j = 4 l1 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]

Helena Bergold and Winfried Hochstättler and Raphael Steiner
2022 arXiv   pre-print
, every 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.  ...

### Minimizing the Oriented Diameter of a Planar Graph

Nicole Eggemann, Steven D. Noble
2009 Electronic Notes in Discrete Mathematics
We consider the problem 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.  ...

### On universal graphs for planar oriented graphs of a given girth

O.V. Borodin, A.V. Kostochka, J. Nešetřil, A. Raspaud, E. Sopena
1998 Discrete Mathematics
We prove here that (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 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.  ...

### (E)-3-(2-Nitrophenyl)-1-{1-phenylsulfonyl-2-[(phenylsulfonyl)methyl]-1H-indol-3-yl}prop-2-en-1-one

S. Karthikeyan, K. Sethusankar, Ganesan Gobi Rajeswaran, Arasambattu K. Mohanakrishnan
2011 Acta Crystallographica Section E
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.  ...  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.  ...

### Oriented coloring of triangle-free planar graphs and 2-outerplanar graphs

Pascal Ochem, Alexandre Pinlou
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.  ...

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

Pascal Ochem, Alexandre Pinlou
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]

Geoffrey R. Grimmett, Zhongyang Li
2015 arXiv   pre-print
The 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)

Can Kocabıyık, Hümeyra Paşaoğlu, Taşkın Basılı, Erbil Ağar
2012 Acta Crystallographica Section E
In the title compound, C 6 H 6 N 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

M. Danish, Masood Hamid, M. Nawaz Tahir, Nazir Ahmad, Sabiha Ghafoor
2010 Acta Crystallographica Section E
The authors acknowledge the provision 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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