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A graph is said to be edge-critical if its chromatic number drops whenever an edge is deleted. Every edge-critical graph is also vertex-critical. The following collection was calculated by Gordon Royle (source) and lists edge-critical graphs on 4 up to 12 vertices.
A fullerene is a cubic planar graph having all faces 5- or 6-cycles. Fullerenes are planar and polyhedral, and every fullerene has exactly twelve 5-cycles. The following collection was calculated by the House of Graphs (source) and list all fullerenes on up to 90 vertices.
Maximal triangle-free graphs
A triangle-free graph is an undirected connected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number 2, graphs with girth 4 (or 0), graphs with no induced 3-cycle, or locally independent graphs.
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A planar graph is a graph that can be embedded in the plane, i.e. it can be drawn on the plane in such a way that its edges intersect only at their endpoints. The following collection was calculated by Brendan McKay (source) and list all non-isomorphic connected planar graphs on up to 9 vertices....
A graph is said to be vertex-critical if its chromatic number drops whenever a vertex is deleted. The following collection was calculated by Gordon Royle (source) and lists vertex-critical graphs on 4 up to 11 vertices.