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**1 - 4**of**4**### An Ore-type Condition for Cyclability

, 2001

"... A graph G is said to be cyclable if for each orientation D of G, there exists a set S(D) ⊆ V (G) such that reversing all the arcs with one end in S results in a Hamiltonian digraph. Let G be a simple graph of even order n ≥ 8. In this paper, we show that if the degree sum of any two nonadjacent ver ..."

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A graph G is said to be cyclable if for each orientation D of G, there exists a set S(D) ⊆ V (G) such that reversing all the arcs with one end in S results in a Hamiltonian digraph. Let G be a simple graph of even order n ≥ 8. In this paper, we show that if the degree sum of any two nonadjacent vertices is not less than n + 1, then G is cyclable and the lower bound is sharp.

### Egalitarian Graph Orientations Glencora Borradaile∗

, 2014

"... Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in pro ..."

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Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order of the indegrees is NP-hard when the resulting orientation is required to be acyclic.

### Neighborhood unions and cyclability of graphs

, 2004

"... A graph G is said to be cyclable if for each orientation G of G, there exists a set S of vertices such that reversing all the arcs of G with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n ¿ 36. In this paper, we show that if for any three independent vertices ..."

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A graph G is said to be cyclable if for each orientation G of G, there exists a set S of vertices such that reversing all the arcs of G with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n ¿ 36. In this paper, we show that if for any three independent vertices x1, x2 and x3, |N (x1) ∪ N (x2)|+ |N (x2) ∪ N (x3)|+ |N (x3) ∪ N (x1)| ¿ 2n+ 1, then G is cyclable.