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3-Vertices with fewest 2-neighbors in plane graphs with no long paths of 2-vertices
Affiliation:1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia;2. Ammosov North-Eastern Federal University, Yakutsk, 677013, Russia
Abstract:Let g(k,t) be the minimum integer such that every plane graph with girth g at least g(k,t), minimum degree δ=2 and no (k+1)-paths consisting of vertices of degree 2, where k1, has a 3-vertex with at least t neighbors of degree 2, where 1t3.In 2015, Jendrol' and Maceková proved g(1,1)7. Later on, Hudák et al. established g(1,3)=10, Jendrol', Maceková, Montassier, and Soták proved g(1,1)7, g(1,2)=8 and g(2,2)11, and we recently proved that g(2,2)=11 and g(2,3)=14.Thus g(k,t) is already known for k=1 and all t. In this paper, we prove that g(k,1)=3k+4, g(k,2)=3k+5, and g(k,3)=3k+8 whenever k2.
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