Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

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Abstract

A fan Fk is a graph that consists of an induced path on k vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers q and k, every graph with sufficiently large chromatic number contains either a clique of size q or a vertex-minor isomorphic to Fk. We also prove that for all positive integers q and k3, every graph with sufficiently large chromatic number contains either a clique of size q or a pivot-minor isomorphic to a cycle of length k.

Keywords

Chromatic number
Chi-bounded class
Vertex-minor
Pivot-mino

Cited by (0)

1

Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2015R1C1A1A02036398).

2

Supported by ERC Starting Grant PARAMTIGHT (No. 280152), and supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC consolidator grant DISTRUCT, agreement No. 648527).

3

Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2011-0011653).