proposed

approved

editing

proposed

A054548 counts graphs covering n vertices with k edges, with loops A369199.


Cf. A014068, A054548, A116508, A133686, A140638, A369143, A369201.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];

Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n], {2}], {n}], Union@@#==Range[n]&&Length[csm[#]]<=1&]]], {n, 0, 5}] (* Gus Wiseman, Feb 14 2024 *)

For at most one cycle we have A005703, labeled A129271, complement A140637.

Cf. A236570 (number of not necessarily connected n-node unicyclic graphs).

Covering case of A005703 (which is also with any number of edges), labeled A129271, complement A140637.

This is the connected case of A137917 and A236570.

A001349 counts unlabeled connected graphs.


Cf. A014068, A116508, A133686, A140638, `A369143, `A369144, `A369201.

Connected This is the connected case of A137917.


Cf. ~A000612, A014068, ~A053530, A116508, `A133686, `A140638, ~A368601, A368835, ~A369141, `A369143, `A369144, ~A369145, ~A369146, `A369201.

Also unlabeled connected choosable simple graphs with n vertices and n edges, where a graph is choosable iff it . The labeled version is possible to choose a different vertex from each edgeA057500. - Gus Wiseman, Feb 12 2024

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];

Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n], {2}], {n}], Union@@#==Range[n]&&Length[csm[#]]<=1&]]], {n, 0, 5}] (* Gus Wiseman, Feb 14 2024 *)

A000088 counts A001434 and A006649 count unlabeled graphs, choosable A134964 with # vertices = # edges.

A001434 and A006649 count unlabeled graphs with #vertices = #edges.

Also unlabeled connected choosable simple graphs with n vertices and n edges, where a graph is choosable iff it is possible to choose a different vertex from each edge. - Gus Wiseman, Feb 12 2024

From Gus Wiseman, Feb 12 2024: (Start)

Representatives of the a(3) = 1 through a(6) = 13 simple graphs:

{12,13,23} {12,13,14,23} {12,13,14,15,23} {12,13,14,15,16,23}

{12,13,24,34} {12,13,14,23,25} {12,13,14,15,23,26}

{12,13,14,23,45} {12,13,14,15,23,46}

{12,13,14,25,35} {12,13,14,15,26,36}

{12,13,24,35,45} {12,13,14,23,25,36}

{12,13,14,23,25,46}

{12,13,14,23,45,46}

{12,13,14,23,45,56}

{12,13,14,25,26,35}

{12,13,14,25,35,46}

{12,13,14,25,35,56}

{12,13,14,25,36,56}

{12,13,24,35,46,56}

(End)

Covering case of A005703 (which is also with any number of edges), labeled A129271, complement A140637.

The labeled version is A057500, connected case of A137916.

Connected case of A137917.

Row k = 1 of A137918.

The version with loops is A368983.

A000088 counts unlabeled graphs, choosable A134964.

A001434 and A006649 count unlabeled graphs with #vertices = #edges.

A006129 counts covering graphs, unlabeled A002494.

A054548 counts graphs covering n vertices with k edges, with loops A369199.


Cf. ~A000612, A014068, ~A053530, A116508, `A133686, `A140638, ~A368601, A368835, ~A369141, `A369143, `A369144, ~A369145, ~A369146, A369201.

approved

editing

More terms from R. Ronald C. Read (rcread(AT)math.uwaterloo.ca)