In several data center settings, each arriving job may only be served by one of a subset of servers. Such a graph constraint can arise due to several reasons. One is locality of the data needed by a job; for example, in content farms (e.g., in Netflix or YouTube) a video request can only be served by a machine that possesses a copy. Motivated by this, we consider a setting where each job, on arrival, reveals a deadline and a subset of servers that can serve it. The job needs to be immediately allocated to one of these servers, and cannot be moved thereafter. Our objective is to maximize the fraction of jobs that are served before their deadlines. For this online load balancing problem, we prove an upper bound of 1-1/e on the competitive ratio of nonpreemptive online algorithms for systems with a large number of servers. We propose an algorithm - INSERT RANKING - which achieves this upper bound. The algorithm makes decisions in a correlated random way and it is inspired by the work of Karp, Vazirani, and Vazirani on online matching for bipartite graphs. We also show that two more natural algorithms, based on independent randomness, are strictly suboptimal, with a competitive ratio of 1/2.