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arXiv · 1502.03146

Increasing paths in edge-ordered graphs: the hypercube and random graphs

Jessica De Silva·Theodore Molla·Florian Pfender·Troy Retter·Michael Tait
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Abstract

An edge-ordering of a graph $G=(V,E)$ is a bijection $\phi:E\to\{1,2,...,|E|\}$. Given an edge-ordering, a sequence of edges $P=e_1,e_2,...,e_k$ is an increasing path if it is a path in $G$ which satisfies $\phi(e_i)<\phi(e_j)$ for all $i<j$. For a graph $G$, let $f(G)$ be the largest integer $\ell$ such that every edge-ordering of $G$ contains an increasing path of length $\ell$. The parameter $f(G)$ was first studied for $G=K_n$ and has subsequently been studied for other families of graphs. This paper gives bounds on $f$ for the hypercube and the random graph $G(n,p)$.

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Jessica De Silva, Theodore Molla, Florian Pfender, Troy Retter, Michael Tait. 2015-02-10. Increasing paths in edge-ordered graphs: the hypercube and random graphs. https://arxiv.org/abs/1502.03146

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