A363043 Triangle read by rows: T(n,k) is the number of unlabeled graphs with n nodes and packing chromatic number k, 1 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 6, 15, 11, 1, 1, 10, 42, 73, 29, 1, 1, 14, 109, 390, 439, 90, 1, 1, 21, 278, 1953, 5546, 4188, 358, 1, 1, 29, 687, 9085, 61023, 134183, 67888, 1771, 1, 1, 41, 1694, 40344, 572235, 3517101, 5860434, 2001582, 11735, 1

Offset: 1

Author

Pontus von Brömssen, May 14 2023

Keywords

• nonn
• tabl

Comments

The concept of the packing chromatic number was introduced by Goddard et al. (2008) under the name broadcast chromatic number. The term packing chromatic number was introduced by Brešar et al. (2007).

Examples

			Triangle begins:
  n\k| 1  2    3     4      5       6       7       8     9 10
  ---+--------------------------------------------------------
   1 | 1
   2 | 1  1
   3 | 1  2    1
   4 | 1  4    5     1
   5 | 1  6   15    11      1
   6 | 1 10   42    73     29       1
   7 | 1 14  109   390    439      90       1
   8 | 1 21  278  1953   5546    4188     358       1
   9 | 1 29  687  9085  61023  134183   67888    1771     1
  10 | 1 41 1694 40344 572235 3517101 5860434 2001582 11735  1
		

Links

• Boštjan Brešar, Sandi Klavžar, and Douglas F. Rall, On the packing chromatic number of Cartesian products, hexagonal lattice, and trees, Discrete Applied Mathematics 155 (2007), 2303-2311.
• Wayne Goddard, Sandra M. Hedetniemi, Stephen T. Hedetniemi, John M. Harris, and Douglas F. Rall, Broadcast chromatic numbers of graphs, Ars Combinatoria 86 (2008), 33-49.

Crossrefs

Cf. A000041, A000088 (row sums), A084268 (chromatic number), A275622 (cubic graphs), A335203 (hypercube graph) A362580 (square grid graph), A363044 (connected).

Formula

T(n,1) = 1. (The only graphs with packing chromatic number 1 are the graphs with no edges.)

T(n,2) = A000041(n)-1. (The only graphs with packing chromatic number 2 are those consisting of star graph components, with at least one of the components containing more than one node.)

T(n,n) = 1. (The only graph with n nodes and packing chromatic number n is the complete graph on n nodes.)