9 research outputs found
Long monotone trails in random edge-labelings of random graphs
Given a graph and a bijection , we say that a trail/path in is -\emph{increasing} if the
labels of consecutive edges of this trail/path form an increasing sequence.
More than 40 years ago Chv\'atal and Koml\'os raised the question of providing
the worst-case estimates of the length of the longest increasing trail/path
over all edge orderings of . The case of a trail was resolved by Graham
and Kleitman, who proved that the answer is , and the case of a path is
still widely open. Recently Lavrov and Loh proposed to study the average case
of this problem in which the edge ordering is chosen uniformly at random. They
conjectured (and it was proved by Martinsson) that such an ordering with high
probability (whp) contains an increasing Hamilton path.
In this paper we consider random graph and its edge ordering
chosen uniformly at random. In this setting we determine whp the asymptotics of
the number of edges in the longest increasing trail. In particular we prove an
average case of the result of Graham and Kleitman, showing that the random edge
ordering of has whp an increasing trail of length and this
is tight. We also obtain an asymptotically tight result for the length of the
longest increasing path for random Erd\H{o}-Renyi graphs with
Nearly-linear monotone paths in edge-ordered graphs
How long a monotone path can one always find in any edge-ordering of the complete graph Kn?
This appealing question was first asked by Chv´atal and Koml´os in 1971, and has since attracted the
attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that
one can always find a monotone path of linear length, but until now the best known lower bound was
n
2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph
contains a monotone path of length n
1−o(1
Nearly-linear monotone paths in edge-ordered graphs
How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n^2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n^1−o(1)