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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 by Little in cite{L73} (cf. cite{LR}) using linear algebra arguments.
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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].
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 labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. An antimagic orientation $D$ of a graph $G$ is $antimagic$ if $D$ has an antimagic labeling. Hefetz, M$ddot{u}$tze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the question: Does every graph admits an antimagic orientation? It had been proved that for any integer $d$, every 2$d$-regular graph with at most two odd components has an antimagic orientation. In this paper, we consider the 2$d$-regular graph with many odd components. We show that every 2$d$-regular graph with any odd components has an antimagic orientation provide each odd component with enough order.
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 orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, M{u}tze and Schwartz. In this paper, we prove that every $2d$-regular graph (not necessarily connected) admits an antimagic orientation, where $dge2$. Together with known results, our main result implies that the above-mentioned conjecture is true for all regular graphs.
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 a minimum degree condition of $(1+c_1)log_2{n}$ and has Cheeger constant at least $c_2frac{log_2log_2{n}}{log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we provide an explicit construction to show our minimum degree condition is tight while the Cheeger constant bound is tight up to a $log_2log_2{n}$ factor.
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 of orientations of $G(n,p)$ containing no strongly connected copy of $K_r$. We give the correct order of growth of $log T_r(n,p)$ and $log S_r(n,p)$ up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.
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