Long path and cycle decompositions of even hypercubes

We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if $n$ is even, $\ell < 2^n$ and $\ell$ divides the number of edges of $Q_n$, then the path of length $\ell$ decomposes $Q_n$. Tapadia et al.\ proved that any path of length $2^mn$, where $2^m<n$, satisfying these conditions decomposes $Q_n$. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to $2^{n+1}/n$ decompose $Q_n$. As a consequence, we show that $Q_n$ can be decomposed into copies of any path of length at most $2^{n}/n$ dividing the number of edges of $Q_n$, thereby settling Erde's conjecture up to a linear factor

Investigations of Protostellar Outflow Launching and Gas Entrainment: Hydrodynamic Simulations and Molecular Emission

We investigate protostellar outflow evolution, gas entrainment, and star formation efficiency using radiation-hydrodynamic simulations of isolated, turbulent low-mass cores. We adopt an X-wind launching model, in which the outflow rate is coupled to the instantaneous protostellar accretion rate and evolution. We vary the outflow collimation angle from $\theta$=0.01-0.1 and find that even well collimated outflows effectively sweep up and entrain significant core mass. The Stage 0 lifetime ranges from 0.14-0.19 Myr, which is similar to the observed Class 0 lifetime. The star formation efficiency of the cores spans 0.41-0.51. In all cases, the outflows drive strong turbulence in the surrounding material. Although the initial core turbulence is purely solenoidal by construction, the simulations converge to approximate equipartition between solenoidal and compressive motions due to a combination of outflow driving and collapse. When compared to a simulation of a cluster of protostars, which is not gravitationally centrally condensed, we find that the outflows drive motions that are mainly solenoidal. The final turbulent velocity dispersion is about twice the initial value of the cores, indicating that an individual outflow is easily able to replenish turbulent motions on sub-parsec scales. We post-process the simulations to produce synthetic molecular line emission maps of $^{12}$CO, $^{13}$CO, and C$^{18}$O and evaluate how well these tracers reproduce the underlying mass and velocity structure.Comment: Accepted to ApJ, 17 pages, 15 figure