How many randomly colored edges make a randomly colored dense graph rainbow Hamiltonian or rainbow connected?

Abstract

In this paper, we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular, we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge‐disjoint rainbow‐Hamilton cycles w.h.p. We also ask when, in the resultant graph, every pair of vertices is connected by a rainbow path w.h.p.

Hamilton Cycles in Random Graphs: a bibliography

“…Anastos and Frieze [10] considered the addition of m randomly colored edges X to a randomly edge colored dense graph H with with minimum degree at least δn. The colors are chosen randomly from [r] and θ = − log δ.…”

Rainbow connectivity of randomly perturbed graphs

“…For r = 3 colors, (i) shows that if m has order of magnitude log n, then G 3 H,m is rainbow connected w.h.p., which shows that (ii) is best possible. Anastos and Frieze [2] put forth that likely part (iii) holds for 5 or 6. In this note we prove the result for r ≥ 5.…”

“…Anastos and Frieze [2] examined random edge-colorings of perturbed graphs. Let G r H,m be the graph G H,m equipped with an r-edge-coloring where the edge-colors are independently and uniformly selected from [r] (in this way the coloring need not be proper).…”

“…Let G r H,m be the graph G H,m equipped with an r-edge-coloring where the edge-colors are independently and uniformly selected from [r] (in this way the coloring need not be proper). In [2] it was shown that for a linear number of edges m and number of colors r ≥ (120 + 20 log δ)n suffice in order for G r H,m to have rainbow-Hamilton cycle. Later Aigner-Horev and Hefetz [1] improved the value of r to the asymptotically best-possible (1 + o(1))n.…”

“…In a perturbed graph G r H,m , Anastos and Frieze [2] proved that if r ≥ 7 and m = ω(1) then with high probability G r H,m is rainbow connected. Moreover, they showed the following.…”