, D has planar embeddings such that all of e, e , f, f have their both sides being incident to the outer face, and, as going along the outer face
, every k-BMRN * -colouring ? of (D, B), we have ?(e) = ?(e ) and ?(f ) = ?(f )
, There exist k-BMRN * -colourings ? of (D, B) where ?(e) = ?(e ) = ?(f ) = ?(f )
, nonplanar matched digraph (D, M ) that we want to k-BMRN * -colour. To ease the following previous arguments on bridges) digraph (D , B ), which is still of girth at least 5 with ? + (B ) ? 1, admits a 7-BMRN * -colouring ?, Also, there exist k-BMRN * -colourings ? of (D, B) where ?(e) = ?(e ) = ?(f ) = ?(f )
, ? the colour of the backbone arc out-going from v
, ? for every interference arc (u, w) out-going from u, the colour of the backbone arc in-coming to w
, ? for every interference arc (w, v) in-coming to v, the colour of the backbone arc outgoing from w
, It can easily be seen that any other arc incident to u or v does not yield any colouring constraint for extending ? to (u, v). Also, any arc incident to u or v constrains the assignation of at most one colour to (u, v)
, Let v be a vertex of (D, B) being adjacent to 1-vertices
, 4. 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 ? + (B) ? 1. The third item and the first part of the fourth item follow from a direct application of Claim 1. Recall in particular that removing a backbone arc from (D, B) results in a connected digraph (since bridges are incident to 1-vertices, by previous arguments), which implies that the girth restriction is preserved upon removing single arcs. 2-vertices u, w. By minimality of (D, B), v must be incident to a backbone arc; assume (u, v) or (v, u) 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, are because every vertex of
, B) has no bad 5-face f = (v 1, Claim, vol.8, issue.5, p.3
, ) and (v 4 , v 4 ) are backbone arcs, while (v 3 , v 2 ), (v 3 , v 4 ) 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. Also, Similarly as in the proof of Claim 7, without loss of generality we may assume that
,
, the arc joining v i and v i yields vertex, then we are done, because either v 1 , v 5 and one of v 1 , v 1 all belong to a same heavy face, or Question 5.1. What is the complexity of Planar 7-BMRN * -Colouring? Our approach of using crossover gadgets is of course still applicable here. However, we were not able to design 7-crossover gadgets. Designing such gadgets indeed requires lots of interference arcs, which hardly comply with the planarity requirement. Nevertheless, our bet is that Planar 7-BMRN * -Colouring should also be NP-hard. Another remaining algorithmic question is about the complexity of Planar k-BMRN-Colouring for planar spanned digraphs, the variant of Planar k-BMRN * -Colouring for BMRNcolouring. Recall that the difference between BMRN-colouring and BMRN * -colouring is that, by the former, it is not mandatory, for every vertex, Similarly as in the proof of Claim 7, without loss of generality we may assume that, vol.5
, the complexity of Planar k-BMRN-Colouring? Note that it could be interesting as well to wonder about these algorithmic concerns for restricted families of planar spanned digraphs. In particular, due to the results in [3], outerplanar spanned digraphs have BMRN * -index (and, thus, BMRN-index) at most 5 (which is tight in general), and the proof of that result implies that the BMRN * -index of such a digraph can be determined in polynomial time. The same, however, was not proved for the BMRN-index of these digraphs
, the complexity of k-BMRN-Colouring when restricted to outerplanar spanned digraphs? Regarding planar backboned digraphs with large girth, we were not able to exhibit some better girth threshold above which 5-BMRN * -colourings always exist. For instance, we think the following question could be an interesting first step to consider: Question 5.4. Is it true that every planar backboned digraph (D, B) with girth at least 11 has BMRN * -index at most 5?
, Another related aspect is how much can the girth conditions in Theorem 4.1 be lowered, namely for a given k ? {3, 4, 5, 6, 7}, what is the smallest g(k) such that planar backboned digraphs (D, B) with g(D) ? g(k) have BMRN * -index at most k
, 7}, what is the smallest g(k) such that planar backboned digraphs (D, B) with g(D) ? g(k) have BMRN * -index at most k? References
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