Regular colouring defect of a cubic graph and the conjectures of Fan-Raspaud and Fulkerson

2023/12/21 by Karabáš, Ján, Máčajová, Edita, Nedela, Roman +1
#05C15 #05C70 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.13638

Abstract

We introduce a new invariant of a cubic graph - its regular colouring defect - which is defined as the smallest number of edges left uncovered by any collection of three perfect matchings that have no edge in common. This invariant is a modification of colouring defect, an invariant introduced by Steffen (J. Graph Theory 78(2015), 195--206), whose definition does not require the empty intersection condition. In this paper we discuss the relationship of this invariant to the well-known conjectures of Fulkerson (1971) and Fan and Raspaud (1994) and prove that colouring defect and regular colouring defect can be arbitrarily far apart.

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