e onsider plne grphs with lrge enough girth gD miniE mum degree δ t lest P nd no (k + 1)Epths onsisting of verties of degree PD where k ≥ 1F sn PHITD rud¡ kD wekov¡ D wdrsD nd § irozki studied the se k = 1D whih mens tht no two PEverties re djentD nd provedD in prtiulrD tht there is QEvertex whose ll three neighors hve degree P @lled soft QEstrAD provided tht g ≥ 10D whih ound on g is shrpF por the (rst open se k = 2 it ws known tht soft QEstr exists if g ≥ 14 ut my not exist if g ≤ 12F sn this

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... pperD we settle the se k = 2 y presenting onstrution with g = 13 nd no soft QEstrF por ll k ≥ 3D we prove tht soft QEstrs exist if g ≥ 4k + 6 utD s follows from our onstrutionD possily not exist if g ≤ 3k + 7F e onjeture tht in ft soft QEstrs exist whenever g ≥ 3k + 8F Keywords: plne grphD struture propertiesD girthD tight desriptionD weightD heightD QEstrD soft QEstrF