Jessica De Silva Department of Mathematics University of - PowerPoint PPT Presentation

Increasing Paths Jessica De Silva Jessica De Silva Department of Mathematics University of Nebraska-Lincoln, USA SP Coding School 2015

Increasing Paths Jessica De Silva

Increasing Paths Jessica De Silva

Increasing Paths Jessica De Silva

Increasing Paths Jessica De Silva Increasing path of length 3.

Increasing Paths Fix a graph G . Jessica De Silva

Increasing Paths Fix a graph G . Jessica De Silva Let ϕ be an edge-ordering of G .

Increasing Paths Fix a graph G . Jessica De Silva Let ϕ be an edge-ordering of G . Define P ( G, ϕ ) to be the length of the longest increasing path in G with edge-ordering ϕ .

Increasing Paths Fix a graph G . Jessica De Silva Let ϕ be an edge-ordering of G . Define P ( G, ϕ ) to be the length of the longest increasing path in G with edge-ordering ϕ . Goal is to find: f ( G ) := ϕ an edge-ordering P ( G, ϕ ) min

Increasing Paths Theorem (Graham and Kleitman 1973) Jessica De Silva 1 � √ ≤ f ( K n ) ≤ 3 � 4 n − 3 − 1 4 n. 2

Increasing Paths Theorem (Graham and Kleitman 1973) Jessica De Silva 1 � √ ≤ f ( K n ) ≤ 3 � 4 n − 3 − 1 4 n. 2 Theorem (D., Molla, Pfender, Retter, Tait 2014+) f ( G ( n, p )) ≥ (1 − o (1)) √ n with high probability whenever p ≥ ω ( n ) log n √ n and ω ( n ) → ∞ arbitrarily slowly.

Increasing Paths Jessica De Silva Research conducted with Theodore Molla, Florian Pfender, Troy Retter, and Michael Tait at The Rocky Mountain-Great Plains Graduate Research Workshop in Combinatorics, supported in part by the NSF under Grant No. DMS-1427526. Research also supported in part by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1041000.