, if v is adjacent to exactly two 1-vertices, then d(v) ? 14; moreover, the two backbone arcs joining v and these two 1-vertices have opposite directions

, B) must be incident to a backbone arc, and we have 2-vertices u, w. By minimality of (D, B), v must be incident to a backbone arc; assume (u, v) is that backbone arc. Then we get a contradiction to Claim 1, since both u and v have degree 2. ? From Claim 3, we know that all -threads in (D, B), if there are any, verify ? {1, 2}. Some of these threads are reducible because of Claim 1. The remaining irreducible threads, The first and second items, as well as the last part of the fourth item

, since D has girth at least 5) having most of their vertices being of small degree. More precisely, we say that a 5-face is bad if it has at most one vertex that is not a 3 ? -vertex, and, if that vertex exists, it is a 4-vertex, 5-vertex or weak 7-vertex. In other words, a 5-face is not bad as soon as it has at least two 4 + -vertices, or whenever it has a 6-vertex, non-weak 7-vertex, or any k-vertex, More reducible configurations We focus on 5-faces of D (which cannot be incident to 1-vertices

. +--vertices, including two 6 + -vertices. A face of D is said almost heavy if it contains at least three 4 + -vertices

, B) has no bad 5-face f = (v 1, vol.4

, Let us assume that (v 1 , v 2 ) is a backbone arc. By minimality of (D, B), (v 1 , v 5 ) is an interference arc. By the same arguments, we deduce sequentially that, vol.3

, ) is a backbone arc, in which case (v 2 , v 3 ) must be an interference arc

, ) is a backbone arc, in which case, applying Claim 1 onto (v 1 , v 2 ), we deduce that the remaining five arcs incident to v 2 must be interference arcs directed toward v 2 . We then get a contradiction when applying Claim 1 onto

, For each v i , let us denote by v i its unique neighbour not in f . Assume without loss of generality that (v 1 , v 1 ) is a backbone arc. By minimality of (D, B), we deduce that the two interference arcs incident to v 1 (thus on f ) are directed towards v 5 and v 2 . Again by minimality, we deduce that both (v 2 , v 2 ) and (v 5 , v 5 ) are backbone arcs, and thus that, ) is a backbone arc can dealt with in a very similar way, using symmetric arguments. ? Claim 7

, B) has no bad 5-face f = (v 1, Claim, vol.8, issue.5, p.3

, ) and (v 5 , v 4 ) are interference arcs. Since v 1 is weak, it is adjacent to a 2-vertex v 1 , thus not on f . Also, the arc joining v 1 and v 1 is a backbone arc, since v 1 supports v 1 . Assume (v 1 , v 1 ) is a backbone arc. By Claim 1, at least five of the other six arcs incident to v 1 must be interference arcs out-going from v 1 . If (v 5 , v 1 ) or (v 2 , v 1 ) is a backbone arc, then we get a contradiction by applying Claim 1 onto it. Thus, both (v 1 , v 2 ) and (v 1 , v 5 ) are interference arcs, Similarly as in the proof of Claim 7, without loss of generality we may assume that

. Lastly,

, ) are interference arcs. Also, since v 1 is is a 4-vertex, still by Claim 1 none of the arcs joining v 5 and v 1 , and v 2 and v 1 can be backbone arcs. By minimality of (D, B), we deduce that (v 5 , v 5 ) and (v 2 , v 2 ) must be backbone arcs, while (v 5 , v 1 ) and (v 1 , v 2 ) are interference arcs. Still by minimality of (D, B), the presence of these two arcs imply that the two remaining arcs incident to v 1 must be backbone arcs, one, Claim 9. If (D, B) has a bad 5-face f =

. +--vertices, Again by minimality of (D, B), two of the three remaining arcs incident to v 1 must be backbone arcs, one being directed toward v 1 , one being directed away from v 1 . Let us thus denote (v 1 , v 1 ) and (v 1 , v 1 ) the two backbone arcs incident to v 1 . Without loss of generality, we may assume that (v 1 , v 1 ) is an interference arc, if a 7-BMRN * -colouring of the remaining backboned digraph cannot be extended to, vol.5

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