OFFSET
1,3
COMMENTS
A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices.
Examples of sharpness for all n are given in the Bickle 2021 paper.
The smallest maximum degree over all 2-trees with diameter 2 and n vertices is ceiling(2/3*n).
LINKS
Allan Bickle, k-Paths of k-Trees, Springer PROMS 388 (2020) 287-291.
Allan Bickle, Wiener indices of maximal k-degenerate graphs, International Journal of Mathematical Combinatorics 2 (2021) 68-79.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
FORMULA
a(n) = ceiling(2/3*(n-1)) for n>15.
EXAMPLE
For n=5, the graph formed by subdividing one edge of a 4-clique is maximal 2-degenerate with diameter 2, and has maximum degree 3. Thus a(5) = 3.
CROSSREFS
KEYWORD
nonn
AUTHOR
Allan Bickle, Oct 13 2022
STATUS
approved