Saturation of edge-ordered graphs

For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum number of edges of a $G$-saturated graph. In particular, we study the order of magnitude of these functions. For (unordered) graphs, $0$-$1$ matrices, and vertex-ordered graphs it was possible to show that the saturation functions are either $O(1)$ or $\Theta(n)$. We show that the saturation functions of edge-ordered graphs are also either $O(1)$ or $\Omega(n)$. However, by finding edge-ordered graphs whose saturation functions are superlinear, we show that such a dichotomy result does not hold in general. Additionally, we consider the semisaturation problem of edge-ordered graphs, a variant of the saturation problem where we do not require that $H$ is $G$-free. We show a general upper bound $O(n \log n)$ and characterize edge-ordered graphs with bounded semisaturation function. We also present various classes of graphs with bounded, linear and superlinear (semi)saturation functions. Along the way, we define a natural variant of the above problem, where the new edge must get the smallest label. The behaviour of the two variants shows many similarities, which motivated us to investigate the second variant extensively as well.