An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph H , the rainbow Turán number ex(n,H) is defined as themaximumnumber of edges in a properly edge-colored graph on n vertices with no rainbow copy of H . We study the rainbow Turán number of even cycles, and prove that for every fixed ε > 0, there is a constant C(ε) such that every properly edge-...

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f(n, H) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H . The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of Kn with rb(n, H) number of colors contains a rainbow copy o...

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

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 > 4, every edge-colored graph with minimum color degree at least k contains a rainbow matching of size at least ⌈k/2⌉. We prove the slightly weaker statement that a rainbow matching...

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)-s...

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ − →rc(T ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow k-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of kth diameter 2 has rainbow k-co...

The weakly connected domination subdivision number sdγw(G) of a connected graph G is the minimum number of edges which must be subdivided (where each edge can be subdivided at most once) in order to increase the weakly connected domination number. The graph is strongγw-subdivisible if for each edge uv ∈ E(G) we have γw(Guv) > γw(G), where Guv is a graph G with subdivided edge uv. The graph is s...