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A graph G is 1-extendable if every edge belongs to at least one 1-factor. Let G be a graph with a 1-factor F. Then an even F-orientation of G is an orientation in which each F-alternating cycle has exactly an even number of edges directed in the same fixed direction around the cycle. In this paper, we examine the structure of 1-extendible graphs G which have no even F-orientation where F is a fixed 1-factor of G. In the case of cubic graphs we give a characterization. In a companion paper [M. Abreu, D. Labbate and J. Sheehan. Even orientations of graphs: Part II], we complete this characterization in the case of regular graphs, graphs of connectivity at least four and k--regular graphs for $kge3$. Moreover, we will point out a relationship between our results on even orientations and Pfaffian graphs developed in [M. Abreu, D. Labbate and J. Sheehan. Even orientations and Pfaffian graphs].
We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little cite{FL}. Our graph theoretical characterization is equivalent to the one proved
A $labeling$ of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to ${1,2,ldots,m}$. A labeling of $D$ is $antimagic$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u in V(D)$ for a labelin
Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, M{u}tze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orien
We count orientations of $G(n,p)$ avoiding certain classes of oriented graphs. In particular, we study $T_r(n,p)$, the number of orientations of the binomial random graph $G(n,p)$ in which every copy of $K_r$ is transitive, and $S_r(n,p)$, the number
We establish mild conditions under which a possibly irregular, sparse graph $G$ has many strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies