Saturation of Edge-Ordered Graphs

For an edge-ordered graph $G$, an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with an arbitrary label to $H$ creates a copy of $G$. The saturation function is the minimum number of edges in a $G$-saturated graph. For (unordered) graphs, $0$-$1$ matrices, and vertex-ordered graphs, the saturation function is always $O(n)$ and satisfies a dichotomy: it is either $O(1)$ or $\Theta(n)$. The saturation function of an edge-ordered graph follows a weaker dichotomy, being either $O(1)$ or $\Omega(n)$. However, by finding edge-ordered graphs whose saturation functions are $\Omega(n \sqrt{\log n})$, we show that $O(n)$ is not a universal upper bound. We also study the semisaturation problem for edge-ordered graphs, a variant of the saturation problem in which $H$ is not required to be $G$-free. We prove a general upper bound $O(n \log n)$ and characterize edge-ordered graphs with bounded semisaturation functions. We then present several families of edge-ordered graphs with bounded, linear, and superlinear (semi)saturation functions. We also introduce a natural variant of saturation in which the added edge is required to receive the smallest label. The behaviour of the two variants is similar in many respects, which motivated us to investigate the second variant extensively.