Subdivisions of oriented cycles in digraphs with large chromatic number

An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any C a cycle with two blocks, every strongly connected di...

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Bibliographic Details
Published inJournal of graph theory Vol. 89; no. 4; pp. 439 - 456
Main Authors Cohen, Nathann, Havet, Frédéric, Lochet, William, Nisse, Nicolas
Format Journal Article
LanguageEnglish
Published Hoboken Wiley Subscription Services, Inc 01.12.2018
Wiley
Subjects
coloring
Combinatorics
Computational Complexity
Computer Science
Data Structures and Algorithms
digraph
Graph theory
Mathematics
subdivision
Subdivisions
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Summary:An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any C a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of C. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out‐degree 2 and two vertices have in‐degree 2).
ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.22360