Inspired by a 1987 result of Hanson and Toft [Edge‐colored saturated graphs, J Graph Theory 11 (1987), 191–196] and several recent results, we consider the following saturation problem for edge‐colored graphs. An edge‐coloring of a graph F is rainbow if every edge of F receives a different color. Let R(F) denote the set of rainbow‐colored copies of F. A t‐edge‐colored graph G is (R(F),t)‐saturated if G does not contain a rainbow copy of F but for any edge e∈E(G¯) and any color i∈[t], the addition of e to G in color i creates a rainbow copy of F. Let sat tfalse(n,frakturR(F)false) denote the minimum number of edges in an (R(F),t)‐saturated graph of order n. We call this the rainbow saturation number of F. In this article, we prove several results about rainbow saturation numbers of graphs. In stark contrast with the related problem for monochromatic subgraphs, wherein the saturation is always linear in n, we prove that rainbow saturation numbers have a variety of different orders of growth. For instance, the rainbow saturation number of the complete graph Kn lies between nlogn/loglogn and nlogn, the rainbow saturation number of an n‐vertex star is quadratic in n, and the rainbow saturation number of any tree that is not a star is at most linear.

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Given a graph H, a graphic sequence π is potentially H-graphic if there is some realization of π that contains H as a subgraph. In 1991, Erdős, Jacobson and Lehel posed the following question:Determine the minimum even integer σ(H, n) such that every n-term graphic sequence with sum at least σ(H, n) is potentially H-graphic.This problem can be viewed as a "potential" degree sequence relaxation of the (forcible) Turán problems. While the exact value of σ(H, n) has been determined for a number of specific classes of graphs (including cliques, cycles, complete bigraphs and others), very little is known about the parameter for arbitrary H. In this paper, we determine σ(H, n) asymptotically for all H, thereby providing an Erdős-Stone-Simonovits-type theorem for the Erdős-Jacobson-Lehel problem.

Let G be a fixed graph and let F be a family of graphs. A subgraph J of G is F-saturated if no member of F is a subgraph of J, but for any edge e in E(G) − E(J), some element of F is a subgraph of J + e. We let ex(F, G) and sat(F, G) denote the maximum and minimum size of an F-saturated subgraph of G, respectively. If no element of F is a subgraph of G, then sat(In this paper, for k ≥ 3 and n ≥ 100 we determine sat(K 3 , K n k ), where K n k is the complete balanced k-partite graph with partite sets of size n. We also give several families of constructions of K t -saturated subgraphs of K n k for t ≥ 4. Our results and constructions provide an informative contrast to recent results on the edge-density

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex $v$ is the number of different colors on edges incident to $v$. Wang and Li conjectured that for $k\geq 4$, every edge-colored graph with minimum color degree at least $k$ contains a rainbow matching of size at least $\left\lceil k/2 \right\rceil$. We prove the slightly weaker statement that a rainbow matching of size at least $\left\lfloor k/2 \right\rfloor$ is guaranteed. We also give sufficient conditions for a rainbow matching of size at least $\left\lceil k/2 \right\rceil$ that fail to hold only for finitely many exceptions (for each odd $k$).