Identifying whether a degree matrix has an edge-disjoint realization is an NP-hard problem. In comparison, identifying whether a tree degree matrix has an edge-disjoint realization is easier, but the task is still challenging. In 1975, a sufficient condition for the tree degree matrices with three rows has been found, but the condition has not been improved since. This paper contains an essential part of the proof which improves the sufficient condition.
This paper explores the relationship between ethnic fractionalization and social capital between 1990-2005. First, using data from 1990, 1997 and 2005 we test for time differences in the impact of ethnic fractionalization on social capital. Subsequently, we examine U.S. data for evidence consistent with the proposed outcomes in the conflict, contact, or hunker-down theses discussed in Putnam (2007). Putnam (2007) examines what happens to “trust" or “social capital" when individuals of different ethnicity are introduced into social, political and/or economic groups over time. Using an instrumental variable (IV) estimator, we find little evidence of heterogeneity in the impact of ethnic fractionalization on social capital over our period of analysis. In addition, using both fixed effect and IV estimators, we reject the contact hypothesis, but find evidence consistent with the outcomes predicted in both the conflict hypothesis and Putnam’s hunker-down hypothesis, in inter-ethnic relations. Due to data limitations, we are unable to test directly which of these two thesis are more relevant for the U.S experience. However, we provide suggestive evidence in support of the conflict hypothesis over the hunker-down hypothesis. Our results suggest that between 1990-2005, as communities in the U.S became more diverse, there was a tendency for social capital to decline.