Colourings of cubic graphs inducing isomorphic monochromatic subgraphs |
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Authors: | Marién Abreu Jan Goedgebeur Domenico Labbate Giuseppe Mazzuoccolo |
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Institution: | 1. Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Potenza, Italy;2. Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Ghent, Belgium;3. Dipartimento di Informatica, Università degli Studi di Verona, Verona, Italy |
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Abstract: | A -bisection of a bridgeless cubic graph is a -colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most . Ban and Linial Conjectured that every bridgeless cubic graph admits a -bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph with has a -edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban-Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above-mentioned conjectures. Moreover, we prove Ban-Linial's Conjecture for cubic-cycle permutation graphs. As a by-product of studying -edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests. |
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Keywords: | bisection colouring computation cubic graph cycle permutation graph linear forest snark |
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