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Journal Articles The Australasian Journal of Combinatorics Year : 2019

On the 2-edge-coloured chromatic number of grids

Abstract

The oriented (resp. 2-edge-coloured) chromatic number χₒ(G) (resp. χ₂(G)) of an undirected graph G is defined as the maximum oriented (resp. 2-edge-coloured) chromatic number of an orientation (resp. signature) of G. Although the difference between χₒ(G) and χ₂(G) can be arbitrarily large, there are, however, contexts in which these two parameters are quite comparable. We here compare the behaviour of these two parameters in the context of (square) grids. While a series of works has been dedicated to the oriented chromatic number of grids, we are not aware of any work dedicated to their 2-edge-coloured chromatic number. We investigate this throughout this paper. We show that the maximum 2-edge-coloured chromatic number of a grid lies in between 8 and 11. We also focus on 2-row grids and 3-row grids, and exhibit bounds on their 2-edge-coloured chromatic number, some of which are tight. Although our results indicate that the oriented chromatic number and the 2-edge-coloured chromatic number of grids are close in general, they also show that these parameters may differ, even for easy instances.
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Dates and versions

hal-02264958 , version 1 (08-08-2019)
hal-02264958 , version 2 (25-10-2019)

Identifiers

  • HAL Id : hal-02264958 , version 2

Cite

Julien Bensmail. On the 2-edge-coloured chromatic number of grids. The Australasian Journal of Combinatorics, 2019, 75 (3), pp.365-384. ⟨hal-02264958v2⟩
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