A369227 -id:A369227 - cp's OEIS frontend

A348222 Number of uniquely-3-colorable graphs on n vertices.

1, 1, 3, 12, 72, 856, 17018, 531568

Offset: 3

Gordon Royle, Oct 08 2021

A graph is uniquely 3-colorable if there is a unique partition of its vertex set into 3 independent sets. This implies that every proper 3-coloring of the graph has this partition as its set of color classes.

			a(3) = 1 and  a(4) = 1 because the complete graph K3 and K4-e are the only such graphs on 3 and 4 vertices, respectively.

Cf. A369227

a(n) = A369227(n,3). - Eric W. Weisstein, Jan 16 2024

A369223 Number of uniquely colorable simple graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 35, 134, 1183, 21319, 761871

Offset: 0

Eric W. Weisstein, Jan 16 2024

The only disconnected uniquely colorable graphs are the empty graphs on n > 1 nodes.

			n = 1: singleton graph K_1 (1 graph).
n = 2: 2-empty graph, path graph P_2 (2 graphs).
n = 3: 3-empty graph, path graph P_3, triangle graph C_3 = K_3 (3 graphs).
n = 4: 4-empty graph, claw graph K_{1,3}, diamond graph K_{1,1,2} = K_4-e, P_4, square graph C_4, tetrahedral graph K_4 (6 graphs).

Eric Weisstein's World of Mathematics, Simple Graph.
Eric Weisstein's World of Mathematics, Uniquely Colorable Graph.

Cf. A001349.

Cf. A369227 (triangle by chromatic number).

a(n) = Sum_{k=1..n} A369227(n,k).

cp's OEIS Frontend

A348222 Number of uniquely-3-colorable graphs on n vertices.

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