A357501 Length of longest induced cycle in the n X n king graph.

0, 3, 4, 8, 12, 16, 24, 31, 38, 47, 60, 71, 82, 95, 112, 127, 142

Offset: 1

Author

Pontus von Brömssen, Oct 01 2022

Keywords

• nonn
• more

Comments

The largest number of nodes of an induced path (instead of cycle) in the n X n king graph is A000982(n) = ceiling(n^2/2) (Beluhov 2023). - Pontus von Brömssen, Jan 30 2023

Examples

			Longest induced cycles for 6 <= n <= 8:
  . X X X X .   . X X X X X .   . X X X X X X .
  X . . . . X   X . . . . . X   X . . . . . . X
  X . . . . X   X . . X . . X   X . . X X . . X
  X . . . . X   X . X . X . X   X . X . . X . X
  X . . . . X   X . X . X . X   X . . X . X . X
  . X X X X .   X . X . X . X   X . . X . X . X
                . X . . . X .   X . . X . X . X
                                . X X . . . X .
		

References

• Donald E. Knuth, The Art of Computer Programming, Volume 4B, Combinatorial Algorithms, Part 2, Addison-Wesley, 2023. See exercise 7.2.2.1-172 and its solution.

Links

• Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.
• Eric Weisstein's World of Mathematics, King Graph.
• Wikipedia, Induced path

Crossrefs

Cf. A000982, A165143, A357357, A361171.

Formula

From Pontus von Brömssen, Jan 30 2023: (Start)

Beluhov (2023) proves that

a(n) = n^2/2-1 if n == 0 (mod 4) and n >= 8;

a(n) = (n^2-1)/2 if n == 3 (mod 4);

and says that experimental data suggests that perhaps

a(n) = (n^2-5)/2 if n == 1 (mod 4) and n >= 13;

a(n) = n^2/2-3 if n == 2 (mod 4) and n >= 14.

(End)

Extensions

a(9)-a(17) from Andrew Howroyd, Mar 03 2023