Long Monotone Trails in Random Edge-Labellings of Random Graphs

Given a graph $G$ and a bijection $f : E(G)\rightarrow \{1, 2, \ldots,e(G)\}$, we say that a trail/path in $G$ is $f$-\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 $K_n$. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is $n-1$, 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 $G=G(n,p)$ 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 $K_n$ has whp an increasing trail of length $(1-o(1))en$ 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 $p=o(1)$.

[1]  F. Spitzer A Combinatorial Lemma and its Application to Probability Theory , 1956 .

[2]  Raphael Yuster Large Monotone Paths in Graphs with Bounded Degree , 2001, Graphs Comb..

[3]  Raphael Yuster,et al.  Monotone paths in edge-ordered sparse graphs , 2001, Discret. Math..

[4]  R. Graham,et al.  Increasing paths in edge ordered graphs , 1973 .

[5]  Michael Tait,et al.  Increasing Paths in Edge-Ordered Graphs: The Hypercube and Random Graph , 2016, Electron. J. Comb..

[6]  Peter Winkler Puzzled: Solutions and sources , 2014, CACM.

[7]  Kevin G. Milans,et al.  Monotone Paths in Dense Edge-Ordered Graphs , 2015, 1509.02143.

[8]  Po-Shen Loh,et al.  Increasing Hamiltonian paths in random edge orderings , 2016, Random Struct. Algorithms.

[9]  Anders Martinsson,et al.  Most edge‐orderings of Kn have maximal altitude , 2016, Random Struct. Algorithms.

[10]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[11]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[12]  Colin McDiarmid,et al.  Minimal Positions in a Branching Random Walk , 1995 .

[13]  Bruce A. Reed,et al.  On the Variance of the Height of Random Binary Search Trees , 1995, SIAM J. Comput..

[14]  Noga Alon,et al.  Problems and results in extremal combinatorics--I , 2003, Discret. Math..

[15]  Benny Sudakov,et al.  Nearly-linear monotone paths in edge-ordered graphs , 2018, Israel Journal of Mathematics.

[16]  V. Chvatal,et al.  Some Combinatorial Theorems on Monotonicity , 1971, Canadian Mathematical Bulletin.

[17]  A. Robert Calderbank,et al.  Increasing sequences with nonzero block sums and increasing paths in edge-ordered graphs , 1984, Discret. Math..