Planar graphs with few vertices of small degree Original Research Article

Given n > k > 0, Erdös and Griggs introduced ak(n) = minG | deg(v) < k|, where G runs over all simple planar graphs on n vertices. With many constructions and the Euler formula, we determine ak(n) for all n ⩾ nk: (i) ak(n) = 0, if k < 6; (ii) a6(n) = 4, if n is even; a6(n) = 5, if n is odd; (iii) ak(n) = ⌈ ((k − 6)n + 12)/(k − 3) ⌉, if k = 7,8,9,10; (iv) a11(n) = ⌈ (5n + 18)/8 ⌉; (v)ak(n) = ⌈ ((k − 8)n + 16)/(k − 6)⌉, if k ⩾ 12. The cases k ⩽ 6 follow from results of Grünbaum and Motzkin, while the cases k ⩾ 12 and the lower bound for a11(n) are due to West and Will.