There are several ways to generalize graph coloring to signed graphs. Máčajová, Raspaud and Škoviera introduced one of them and conjectured that in this setting, for signed planar graphs four colours are always enough, generalising thereby The Four Color Theorem. We disprove the conjecture.
Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for trianglefree graphs. * M.B., A.L. and J.N. are supported by ANR project GrR (ANR 18 CE40 0032). T. K. is supported by the grant no. 19-04113Y of the Czech Science Foundation (GAČR) and the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004), her visit to Bordeaux, where part of the research was conducted, was supported by Czech-French Mobility project 8J19FR027.1 Initially in the context of vertex colouring.Note that while some graphs need ∆(G) + 1 colours, some graphs can be edge-coloured with only ∆(G) colours. In the follow-up paper extending the result to multigraphs [Viz65], and later in a more publicly available survey paper [Viz68], Vizing asks whether an optimal colouring can always be reached through a series of Kempe changes, as follows.Question 1.3 ([Viz65]). For every simple graph G, for any integer k > χ ′ (G), for any k-edgecolouring α, is there a χ ′ (G)-edge-colouring that can be reached from α through a series of Kempe changes? Question 1.3 is in fact stated in the more general context of multigraphs. Note that neither Theorem 1.2 nor Question 1.3 implies that all colourings with fewer colours are reachable, i.e., there is no choice regarding the target colouring. We say two kedge-colourings are Kempe-equivalent if one can be reached from the other through a series of Kempe changes using colours from {1, . . . , k}. Question 1.3, if true, would imply [AC16] and the following conjecture of Mohar [Moh06], using the target χ ′ (G)-colouring as an intermediate colouring.Conjecture 1.4 ([Moh06]). For every simple graph G, all (∆(G) + 2)-edge-colourings are Kempe-equivalent.Mohar proved the weaker case where (χ ′ (G) + 2) colours are allowed.Theorem 1.5 ([Moh06]). For every simple graph G, all (χ ′ (G) + 2)-edge-colourings are Kempeequivalent.
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