
I will survey some sufficient conditions for a graph $G$ to admit a fractional triangle decomposition. The focus is on graphs which are `locally dense' in the complete graph or complete tripartite graph. Using a recent theorem of Barber et al., this leads to asymptotic results on genuine triangle decompositions.
An application is the completion problem for partial latin squares, a topic investigated in Alex Rosa's work.
In 1966 Rosa introduced a sequence R = $(r_1, r_2, . . . , r_{2n+1} )$ of 2n+1 integers that satisfy the following conditions:
(1) For every $k \in$ $\{1,2, ...,n \}$ there exist exactly two elements $r_i ,r_j$ such that $r_i= r_j = k$. \
(2) If $r_i= r_j= k$ with $i < j$, then $j  i = k$. \
(3) $r_{n+1} = 0 $ \
For example, (1,1,3,4,0,3,2,4,2) is a Rosa sequence of order 4 . \
In this talk, we survey the development of Rosa sequences and reflect on examples of Professor Rosa as a supervisor.