  Online bipartite matching with one-sided arrival and its variants have been
extensively studied since the seminal work of Karp, Vazirani, and Vazirani
(STOC 1990). Motivated by real-life applications with dynamic market
structures, e.g. ride-sharing, two generalizations of the classical one-sided
arrival model are proposed to allow non-bipartite graphs and to allow all
vertices to arrive online. Namely, online matching with general vertex arrival
is introduced by Wang and Wong (ICALP 2015), and fully online matching is
introduced by Huang et al. (JACM 2020).
  In this paper, we study the fractional versions of the two models. We improve
three out of the four state-of-the-art upper and lower bounds of the two
models. For fully online matching, we design a $0.6$-competitive algorithm and
prove no algorithm can be $0.613$-competitive. For online matching with general
vertex arrival, we prove no algorithm can be $0.584$-competitive. Moreover, we
give an arguably more intuitive algorithm for the general vertex arrival model,
compared to the algorithm of Wang and Wong, while attaining the same
competitive ratio of $0.526$.
