Induced subgraphs of graphs with large chromatic number IX: Rainbow paths

Alex Scott; Paul Seymour

doi:10.37236/6768

Induced subgraphs of graphs with large chromatic number IX: Rainbow paths

Alex Scott
, Paul Seymour

Mathematics

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

7 Scopus citations

Abstract

We prove that for all integers k, s ≥ 0 there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016), who proved the same for graphs G with k = 2 and girth at least five.

Original language	English (US)
Article number	#P2.53
Journal	Electronic Journal of Combinatorics
Volume	24
Issue number	2
DOIs	https://doi.org/10.37236/6768
State	Published - Jun 30 2017

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

Keywords

Colouring
Rainbow

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10.37236/6768

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abstract = "We prove that for all integers k, s ≥ 0 there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gy{\'a}rf{\'a}s and S{\'a}rk{\"o}zy (2016), who proved the same for graphs G with k = 2 and girth at least five.",

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AB - We prove that for all integers k, s ≥ 0 there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016), who proved the same for graphs G with k = 2 and girth at least five.

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ER -

Induced subgraphs of graphs with large chromatic number IX: Rainbow paths

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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