TY - JOUR

T1 - Compatible Hamilton cycles in random graphs

AU - Krivelevich, Michael

AU - Lee, Choongbum

AU - Sudakov, Benny

N1 - Publisher Copyright:
© 2016 Wiley Periodicals, Inc.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability p(n) » log n/n, the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph (Formula presented.), an incompatibility system F over G is a family F = {Fv}v∈V where for every v∈V, the set Fv is a set of unordered pairs p(n) » log n/n. An incompatibility system is Δ-bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in Fv containing e. We say that a cycle C in G is compatible with F if every pair of incident edges e, e' of C satisfies {e, e'}. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant μ > 0 such that the random graph G = G(n, p) with F is asymptotically almost surely such that for any μnp-bounded incompatibility system F over G, there is a Hamilton cycle in G compatible with F. We also prove that for larger edge probabilities p(n) » log n/n, the parameter μ can be taken to be any constant smaller than 1-1/√2. These results imply in particular that typically in G(n, p) for p » log n/n, for any edge-coloring in which each color appears at most μnp times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge-coloring of such a random graph in which each color appears on at most μnp edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle).

AB - A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability p(n) » log n/n, the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph (Formula presented.), an incompatibility system F over G is a family F = {Fv}v∈V where for every v∈V, the set Fv is a set of unordered pairs p(n) » log n/n. An incompatibility system is Δ-bounded if for every vertex v and an edge e incident to v, there are at most Δ pairs in Fv containing e. We say that a cycle C in G is compatible with F if every pair of incident edges e, e' of C satisfies {e, e'}. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant μ > 0 such that the random graph G = G(n, p) with F is asymptotically almost surely such that for any μnp-bounded incompatibility system F over G, there is a Hamilton cycle in G compatible with F. We also prove that for larger edge probabilities p(n) » log n/n, the parameter μ can be taken to be any constant smaller than 1-1/√2. These results imply in particular that typically in G(n, p) for p » log n/n, for any edge-coloring in which each color appears at most μnp times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge-coloring of such a random graph in which each color appears on at most μnp edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle).

KW - Hamilton cycles

KW - incomparability systems

KW - rainbow cycles

KW - random graphs

UR - http://www.scopus.com/inward/record.url?scp=84981293954&partnerID=8YFLogxK

U2 - 10.1002/rsa.20636

DO - 10.1002/rsa.20636

M3 - מאמר

AN - SCOPUS:84981293954

VL - 49

SP - 533

EP - 557

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -