Vertex Degrees in Outerplanar Graphs

For an outerplanar graph on \( n \) vertices, we determine the maximum number of vertices of degree at least \( k \). For \( k = 4 \) (and \( n \geq 7 \)) the answer is \( n – 4 \). For \( k = 5 \) (and \( n \geq 4 \)), the answer is \( \left\lfloor \frac{2(n-4)}{3} \right\rfloor \) (except one less when \( n \equiv 1 \pmod{6} \)). For \( k \geq 6 \) (and \( n \geq k + 2 \)), the answer is \( \left\lfloor \frac{n-6}{k-4} \right\rfloor \). We also determine the maximum sum of the degrees of \( s \) vertices in an \( n \)-vertex outerplanar graph and the maximum sum of the degrees of the vertices with degree at least \( k \).