Monotone paths in edge-ordered sparse graphs

Yehuda Roditty; Barack Shoham; Raphael Yuster

doi:10.1016/S0012-365X(00)00174-6

Monotone paths in edge-ordered sparse graphs

Yehuda Roditty, Barack Shoham, Raphael Yuster

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

An edge-ordered graph is an ordered pair (G,f), where G = G(V,E) is a graph and f is a bijective function, f : E(G) → {1,2,..., |E(G)|}. f is called an edge ordering of G. A monotone path of length k in (G, f) is a simple path P_k=1: v₁, v₂,..., v_k+1 in G such that either, f((v_i, v_i+1)) < f((V_i+1,V_i+2)) or f((v_iv_i+1)) > f((v_i+1, v_i+2)) for i = 1,2,..., k - 1. Given an undirected graph G, denote by α(G) the minimum over all edge orderings of the maximum length of a monotone path. In this paper we give bounds on α(G) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.

Original language	English
Pages (from-to)	411-417
Number of pages	7
Journal	Discrete Mathematics
Volume	226
Issue number	1-3
DOIs	https://doi.org/10.1016/S0012-365X(00)00174-6
State	Published - 2001

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/S0012-365X(00)00174-6

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title = "Monotone paths in edge-ordered sparse graphs",

abstract = "An edge-ordered graph is an ordered pair (G,f), where G = G(V,E) is a graph and f is a bijective function, f : E(G) → \{1,2,..., |E(G)|\}. f is called an edge ordering of G. A monotone path of length k in (G, f) is a simple path Pk=1: v1, v2,..., vk+1 in G such that either, f((vi, vi+1)) < f((Vi+1,Vi+2)) or f((vivi+1)) > f((vi+1, vi+2)) for i = 1,2,..., k - 1. Given an undirected graph G, denote by α(G) the minimum over all edge orderings of the maximum length of a monotone path. In this paper we give bounds on α(G) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.",

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AU - Yuster, Raphael

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AB - An edge-ordered graph is an ordered pair (G,f), where G = G(V,E) is a graph and f is a bijective function, f : E(G) → {1,2,..., |E(G)|}. f is called an edge ordering of G. A monotone path of length k in (G, f) is a simple path Pk=1: v1, v2,..., vk+1 in G such that either, f((vi, vi+1)) < f((Vi+1,Vi+2)) or f((vivi+1)) > f((vi+1, vi+2)) for i = 1,2,..., k - 1. Given an undirected graph G, denote by α(G) the minimum over all edge orderings of the maximum length of a monotone path. In this paper we give bounds on α(G) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.

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Monotone paths in edge-ordered sparse graphs

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