Equitable tricolourings for 4-cycle systems

LetΣ=(X,B)a 4-cycle system of orderv=1+8k. Ac-colouringof typesis a mapφ∶B→C, withCset of colours, such that exactlyccolours are used and for every vertexxall the blocks containingxare coloured exactly withscolours. Let 4k=qs+r, withq,r≥0.φisequitableif for every vertexxthe set of the 4kblocks containingxisparted inrcolour classes of cardinalityq+1 ands−r colour classes ofcardinalityq. In this paper we study tricolourings, for whichs=3, withthe hypothesis that eitherv≡9 mod 24 orv≡17 mod 24, determiningan upper bound forc.