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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2605.16028 |
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| _version_ | 1866917500277489664 |
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| author | Bernardi, Oliver Fang, Jonathan J. |
| author_facet | Bernardi, Oliver Fang, Jonathan J. |
| contents | A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. Namely, given two sets of pairs of vertices $A=\{(u_i,v_i), i\in[k]\}$ and $B=\{(u_i',v_i'), i\in[l]\}$, we consider the orientations $α$ of $G$ such that adding the elements of $A$ and $B$ as additional directed edges to $α$ gives an orientation $α'$ in which $v_i$ cannot reach $u_i$ for all $i\in[k]$, but $v_i'$ can reach $u_i'$ for all $i\in[l]$. We show that this set of orientations is equinumerous to a set of subgraphs satisfying the ``same" connectivity constraints defined in terms of $A$ and $B$.
We also extend our results to the enumeration of equivalence classes of orientations satisfying such connectivity constraints. Precisely, we consider the equivalence classes under cycle reversal, cocycle reversal, or cycle-cocycle reversal. We show that the equivalences classes are equinumerous to some sets of subgraphs defined by connectivity and acyclicity constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16028 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subgraphs versus Orientations: Infinite families of equidistributions Bernardi, Oliver Fang, Jonathan J. Combinatorics A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. Namely, given two sets of pairs of vertices $A=\{(u_i,v_i), i\in[k]\}$ and $B=\{(u_i',v_i'), i\in[l]\}$, we consider the orientations $α$ of $G$ such that adding the elements of $A$ and $B$ as additional directed edges to $α$ gives an orientation $α'$ in which $v_i$ cannot reach $u_i$ for all $i\in[k]$, but $v_i'$ can reach $u_i'$ for all $i\in[l]$. We show that this set of orientations is equinumerous to a set of subgraphs satisfying the ``same" connectivity constraints defined in terms of $A$ and $B$. We also extend our results to the enumeration of equivalence classes of orientations satisfying such connectivity constraints. Precisely, we consider the equivalence classes under cycle reversal, cocycle reversal, or cycle-cocycle reversal. We show that the equivalences classes are equinumerous to some sets of subgraphs defined by connectivity and acyclicity constraints. |
| title | Subgraphs versus Orientations: Infinite families of equidistributions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.16028 |