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### An Orientation Theorem for Graphs

A.M.H. Gerards
1994 Journal of combinatorial theory. Series B (Print)
We characterize the class of graphs in which the edges can be oriented in such a way that going along any circuit in the graph, the number of forward edges minus the number of backward edges is equal to  ...  The result follows by applying Tutte's characterization of regular matroids to a certain binary matroid associated to a graph. irj  ...  ACKNOWLEDGMENT I thank Andras Frank and Alexander Schrijver for stimulating discussions.  ...

### Simultaneous well-balanced orientations of graphs

Zoltán Király, Zoltán Szigeti
2006 Journal of combinatorial theory. Series B (Print)
Nash-Williams' well-balanced orientation theorem [11] is extended for orienting several graphs simultaneously.  ...  We also have new results about simultaneous well-balanced orientations of non-disjoint subgraphs of an Eulerian graph as well as those of different contractions of a graph.  ...  Acknowledgement We thank András Frank for many helpful discussions.  ...

### Orientations of graphs

Zoltán Szigeti
2010 Matemática contemporânea
We provide here a short survey on orientations of graphs. We concentrate on problems with connectivity properties.  ...  It is known that this is true for Eulerian graphs [1]: Theorem 7.1 (Berg-Jordán).  ...  For every partition {V 1 , . . . V k } of V (G), there exists an orientation G of G such that G and G/V i ∀i, are best-balanced orientations of the corresponding graphs.  ...

### On ordered graphs and graph orderings

Jaroslav Nešetřil
1994 Discrete Applied Mathematics
A graph H is said to have the ordering property for (G, d ) if for every ordered graph (H, 5) (i.e. for every linear ordering 3 of vertices of H) there exists an induced subgraph G' = (V, E') such that  ...  In this note we discuss various existential problems related to graph orientations, to ordered graphs and their relationship to Ramsey theory.  ...  (d): for every ordered graph (G, < ), G E Forb(d) there exists an ordered graph (H, < ), H E Forb(d) such that (H, <) + (G, <)t. (3) Oriented Ramsey Theorem for Forb(d): For every acyclic orientation G  ...

### Oriented graph coloring

Eric Sopena
2001 Discrete Mathematics
An oriented k-coloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset  ...  We survey the main results that have been obtained on oriented graph colorings.  ...  An oriented graph is then nice if it is n-nice for some n. More precisely, we have [17] : Theorem 14 (NeÄ setÄ ril et al. [17] ).  ...

### A dualistic approach to bounding the chromatic number of a graph

Jaroslav Nešetřil, Claude Tardif
2008 European journal of combinatorics (Print)
We exhibit infinitely many examples where this general setting improves the bounds for chromatic number of graphs and we relate this to extremal problems for oriented graphs.  ...  We give a new and more direct proof of the characterization theorem for finitary homomorphism dualities of directed graphs.  ...  This research was done while the second author was visiting the Institute for Theoretical Computer Science (ITI) in the fall of 2000. The support of DIMATIA is gratefully acknowledged.  ...

### A survey on the skew energy of oriented graphs [article]

Xueliang Li, Huishu Lian
2015 arXiv   pre-print
Let G^σ be an oriented graph of G with skew adjacency matrix S(G^σ).  ...  In this paper, we summarize main results on the skew energy of oriented graphs. Some open problems are proposed for further study.  ...  It is almost sure that for every graph G, the energy of G is less than the skew energy of an oriented graph G σ of G. Theorem 9.4.  ...

### Dualities and reciprocities on graphs on surfaces [article]

Woo-Seok Jung, Jaeseong Oh
2021 arXiv   pre-print
We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally  ...  In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations.  ...  In this subsection, we provide a reciprocity theorem for integral local tension as follows. Theorem 4.11. Let G be an oriented graph on a surface.  ...

### Antisymmetric flows and strong colourings of oriented graphs

J. Nešetřill, André Raspaud
1999 Annales de l'Institut Fourier
-Every bridgeless graph G has an orientation for wich G has a Zy-ASF (i.e. asf^(G') < 7 for every bridgeless graph G). Proof. -By Seymour's theorem [25] , any bridgeless graph has a Z2 x Zs-NZF.  ...  We shall give several proofs of the fact that an ASF exists for any oriented graph without oriented 2-cut.  ...

### Eigenvalues of oriented-graph matrices

Jiong-Sheng Li
1995 Linear Algebra and its Applications
We give bounds on the real and imaginary parts of the eigenvalues of an oriented-graph matrix, prove that each of the irreducible oriented-graph matrices of order n 2 3 has at least three distinct eigenvalues  ...  We also consider spectral properties of the bipartite oriented-graph matrices and the multiequipartite oriented-graph matrices.  ...  For the imaginary parts of the eigenvalues of an oriented-graph matrix, we have THEOREM 2.4.  ...

### Orientations of infinite graphs [article]

Marcel Koloschin, Max Pitz
2021 arXiv   pre-print
Building on recent work by Thomassen, we show that Nash-Williams' orientation theorem, that every finite 2k-edge-connected multigraph has a k-arc-connected orientation, also holds for all infinite multigraphs  ...  Nash-Williams' orientation theorem for infinite graphs We are now ready to extend Nash-Williams' orientation theorem to all infinite multigraphs.  ...  Thomassen proved in [12, Theorem 4 ] that for any finite set of vertices A 0 in a 4k-edge-connected locally finite graph G, there is an immersion in G of a finite Eulerian 2k-edge-connected graph on A  ...

### Context Directed Reversals and the Ciliate Decryptome [article]

C.L. Jansen, M. Scheepers, S.L. Simon, E. Tatum
2016 arXiv   pre-print
The cds Rescue Theorem is discussed in the context of a mathematical model for ciliate micronuclear decryption.  ...  cdr Parity Theorem).  ...  Acknowledgements The research represented in this paper was funded by an NSF REU grant DMS 1359425, by Boise State University and by the Department of Mathematics at Boise State University.  ...

### A note on ''The comparability graph of a tree"

E. S. Wolk
1965 Proceedings of the American Mathematical Society
We say that a relation T on G is an orientation of (G, R) if and only if (i) (G, T) is an oriented graph, and (ii) x R y if and only if x T y or y T x, for all x, y EG.  ...  An unoriented graph which possesses a transitive orientation will be called a comparability graph.  ...

### On semi-transitive orientability of Kneser graphs and their complements [article]

Sergey Kitaev, Akira Saito
2019 arXiv   pre-print
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path v_0→ v_1→...→ v_k either there is no edge between v_0 and v_k, or v_i→ v_j is an edge for all 0≤ i<j≤ k.  ...  An undirected graph is semi-transitive if it admits a semi-transitive orientation.  ...  Acknowledgment We are grateful to Li-Da Tong for raising our interest in Kneser graphs.  ...

### On operations preserving semi-transitive orientability of graphs [article]

Ilkyoo Choi, Jinha Kim, Minki Kim
2018 arXiv   pre-print
Moreover, for all three graph operations,we showthat the initial semi-transitive orientation can be extended to the new graph obtained by the graph operation.  ...  We consider the class of semi-transitively orientable graphs, which is a much larger class of graphs compared to transitively orientable graphs, in other words, comparability graphs.  ...  By Theorem 1.11, it is enough to show Theorem 1.12 for edge-deletions. Theorem 2.2. Let e be an edge of a semi-transitively orientable graph G .  ...
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