Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of $T$, then $G$ can be edge-decomposed into subgraphs isomorphic to $T$. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from $T$ by vertex-identifications. We call such a subgraph a homomorphic copy of $T$. This implies the Tree Decomposition Conjecture under the additional constraint that the girth of $G$ is greater than the diameter of $T$. As an application, we verify the Tree Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page

Spanning trees without adjacent vertices of degree 2

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2

Using geographic profiling to locate elusive nocturnal animals: A case study with spectral tarsiers

© 2015 The Zoological Society of London. Estimates of biodiversity, population size, population density and habitat use have important implications for management of both species and habitats, yet are based on census data that can be extremely difficult to collect. Traditional assessment techniques are often limited by time and money and by the difficulties of working in certain habitats, and species become more difficult to find as population size decreases. Particular difficulties arise when studying elusive species with cryptic behaviours. Here, we show how geographic profiling (GP) - a statistical tool originally developed in criminology to prioritize large lists of suspects in cases of serial crime - can be used to address these problems. We ask whether GP can be used to locate sleeping sites of spectral tarsiers Tarsius tarsier in Sulawesi, Southeast Asia, using as input the positions at which tarsier vocalizations were recorded in the field. This novel application of GP is potentially of value as tarsiers are cryptic and nocturnal and can easily be overlooked in habitat assessments (e.g. in dense rainforest). Our results show that GP provides a useful tool for locating sleeping sites of this species, and indeed analysis of a preliminary dataset during field work strongly suggested the presence of a sleeping tree at a previously unknown location; two sleeping trees were subsequently found within 5m of the predicted site. We believe that GP can be successfully applied to locating the nests, dens or roosts of elusive animals such as tarsiers, potentially improving estimates of population size with important implications for management of both species and habitats.We thank Operation Wallacea for supporting S.C.F. in thisproject and for providing logistical support for the fieldwork,and Aidan Kelsey for invaluable assistance in the field. Wethank the Indonesian Institute of Sciences (LIPI) andKementerian Riset dan Teknologi Republik Indonesia(RISTEK) for providing permission to undertake the work(RISTEK permit no. 211/SIP/FRP/SM/VI/2013, and BalaiKonservasi Sumber Daya Alam (BKSDA) for theirassistance