login
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

%I #5 May 14 2023 11:41:27

%S 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,

%T 1,1,21,278,1953,5546,4188,358,1,1,29,687,9085,61023,134183,67888,

%U 1771,1,1,41,1694,40344,572235,3517101,5860434,2001582,11735,1

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

%C 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).

%H Boštjan Brešar, Sandi Klavžar, and Douglas F. Rall, <a href="https://doi.org/10.1016/j.dam.2007.06.008">On the packing chromatic number of Cartesian products, hexagonal lattice, and trees</a>, Discrete Applied Mathematics 155 (2007), 2303-2311.

%H Wayne Goddard, Sandra M. Hedetniemi, Stephen T. Hedetniemi, John M. Harris, and Douglas F. Rall, <a href="https://www.researchgate.net/publication/220620011">Broadcast chromatic numbers of graphs</a>, Ars Combinatoria 86 (2008), 33-49.

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

%F 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.)

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

%e Triangle begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10

%e ---+--------------------------------------------------------

%e 1 | 1

%e 2 | 1 1

%e 3 | 1 2 1

%e 4 | 1 4 5 1

%e 5 | 1 6 15 11 1

%e 6 | 1 10 42 73 29 1

%e 7 | 1 14 109 390 439 90 1

%e 8 | 1 21 278 1953 5546 4188 358 1

%e 9 | 1 29 687 9085 61023 134183 67888 1771 1

%e 10 | 1 41 1694 40344 572235 3517101 5860434 2001582 11735 1

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

%K nonn,tabl

%O 1,5

%A _Pontus von Brömssen_, May 14 2023