A357815 Smallest maximum degree over all maximal 2-degenerate graphs with diameter 2 and n vertices.

0, 1, 2, 3, 3, 4, 4, 4, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 46

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

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

a(n) = ceiling(2/3*(n-1)) for n>15.

G.f.: x^2*(1 + x + x^2 - x^3 - x^5 - x^7 + x^8 + x^9 + x^13 - x^17)/((1 - x)^2*(1 + x + x^2)). - Andrew Howroyd, Nov 23 2025

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

Cf. A004523.

Sequence in context: A225634 A247134 A080604 * A221983 A029118 A129382

Adjacent sequences: A357812 A357813 A357814 * A357816 A357817 A357818

Keyword

nonn,easy

Author

Allan Bickle, Oct 13 2022

Extensions

More terms from Andrew Howroyd, Nov 23 2025

Status

approved