Pattern Colored Hamilton Cycles in Random Graphs

Abstract

We consider the existence of patterned Hamilton cycles in randomly colored random graphs. Given a string Π over a set of colors {1, 2, . . . , r}, we say that a Hamilton cycle is Π-colored if the pattern repeats at intervals of length |Π| as we go around the cycle. We prove a hitting time for the existence of such a cycle. We also prove a hitting time result for a related notion of Π-connected.The next phase of this study, concerns Hamilton cycles in random k-uniform hypergraphs. There are various notions of Hamilton cycle in this context, and Ferber and Krivelevich [7] proved that if m H edges are needed for a given type of hamilton cycle to exist w.h.p. then O(m H ) random edges and (1 + ǫ)m H colors are sufficient for a rainbow Hamilton cycles. The hypergraph results in [7] were sharpened by Dudek, English and Frieze [5]. Cooper and Frieze [4] considered the related question of what is the threshold for every k-bounded coloring of the edges of G n,m to contain at least one rainbow Hamilton cycle.Here k-bounded means that no color can be used more than k times.

Hamilton Cycles in Random Graphs: a bibliography

“…We say that a Hamilton cycle with edges e 1 , e 2 , ..., e n is Π-colored if e j has color Π t , where t = j mod ℓ. It is shown by Anastos and Frieze [9] that w.h.p. the hitting time for the existence of a Π-colored Hamilton cycle coincides with the hitting time for every vertex to fit Π.…”

Color‐biased Hamilton cycles in random graphs

“…The events  P ,  0 ,  𝓁 are mutually independent, hence P( P ∧  0 ∧  𝓁 ) ≤ p 𝓁 n −1.2 . Let  be the event that there exists a path P = v 0 , … , v 𝓁 with 𝓁 ∈ [4] in G such that  P and 𝑑(v 0 ), 𝑑(v 𝓁 ) ≤ 𝛼 log n. By the union bound,…”

“…Several concrete manifestations of this phenomenon have been demonstrated in prior works. For example, it is known that the number of Hamilton cycles in G(n, p) is-in some well-defined quantitative sense-concentrated around its mean [20]; that the set of Hamilton cycles in G(n, p) typically possesses local resilience properties [29,33,34,38]; and that random edge-colorings of G(n, p) typically admit Hamilton cycles colored according to any prescribed pattern [4,15].…”

The Threshold for the Full Perfect Matching Color Profile in a Random Coloring of Random Graphs

“…Randomly colored random graphs have been extensively studied in various contexts throughout the last two decades. A few examples include (i) rainbow spanning graphs such as matchings and Hamilton cycles, see e.g., [2], [8], [10], [11], [14]; (ii) rainbow connection, see e.g., [4], [13], [15], [16]; (iii) pattern colored Hamilton cycles, see e.g., [1], [5], [12]; (iv) packing problems, see e.g., [9]. Continuing the research in this line, Frieze defined an elegant notion of a color profile in [6] and gave bounds on the matching color profile for randomly colored random bipartite graphs.…”