One-Way Trail Orientations

Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins\u27 theorem [Robbins, Am. Math. Monthly, 1939] asserts that such an orientation exists if and only if the graph is 2-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph are partitioned into trails. Can the trails be oriented consistently such that the resulting directed graph is strongly connected? We show that 2-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation. The generalised Robbins\u27 theorem [Boesch, Am. Math. Monthly, 1980] for mixed multigraphs asserts that the undirected edges of a mixed multigraph can be oriented to make the resulting directed graph strongly connected exactly when the mixed graph is strongly connected and the underlying graph is bridgeless. We consider the natural extension where the undirected edges of a mixed multigraph are partitioned into trails. It turns out that in this case the condition of the generalised Robbin\u27s Theorem is not sufficient. However, we show that as long as each cut either contains at least 2 undirected edges or directed edges in both directions, there exists an orientation of the trails such that the resulting directed graph is strongly connected. Moreover, if the condition is satisfied, we may start by orienting an arbitrary trail in an arbitrary direction. Using this result one obtains a very simple polynomial time algorithm for finding a strong trail orientation if it exists, both in the undirected and the mixed setting

The Return of Activity in Main-Belt Comet 133P/Elst-Pizarro

Comet 133P/Elst-Pizarro is the first-known and currently best-characterised member of the main-belt comets, a recently-identified class of objects that exhibit cometary activity but which are dynamically indistinguishable from main-belt asteroids. We report here on the results of a multi-year monitoring campaign from 2003 to 2008, and present observations of the return of activity in 2007. We find a pattern of activity consistent with the seasonal activity modulation hypothesis proposed by Hsieh et al. (2004, AJ, 127, 2997). Additionally, recomputation of phase function parameters using data in which 133P was inactive yields new IAU parameters of H_R=15.49+/-0.05 mag and G_R=0.04+/-0.05, and linear parameters of m_R(1,1,0)=15.80+/-0.05 mag and Beta=0.041+/-0.005 mag/deg. Comparison between predicted magnitudes using these new parameters and the comet's actual brightnesses during its 2002 and 2007 active periods reveals the presence of unresolved coma during both episodes, on the order of ~0.20 of the nucleus cross-section in 2002 and ~0.25 in 2007. Multifilter observations during 133P's 2007 active outburst yield mean nucleus colours of B-V=0.65+/-0.03 mag, V-R=0.36+/-0.01 mag, and R-I=0.32+/-0.01 mag, with no indication of significant rotational variation, and similar colours for the trail. Finally, while 133P's trail appears shorter and weaker in 2007 than in 2002, other measures of activity strength such as dust velocity and coma contamination of nucleus photometry are found to remain approximately constant. We attribute changes in trail strength to the timing of observations and projection effects, thus finding no evidence of any substantial decrease in activity strength between 2002 and 2007.Comment: 31 pages, 13 figures, accepted for publication in MNRA