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Unexpectedly, a symmetry on unlabeled graphs

Florian Fürnsinn, Moritz Gangl, Martin Rubey

TL;DR

The paper addresses the problem of finding a natural, nontrivial equidistribution of two graph parameters—tuft number $t(G)$ and sibling number $s(G)$—on unlabeled connected graphs with a fixed number of vertices. It develops a comprehensive, species-theoretic framework, introducing co-mating graphs, 3-sort graphs, tufts, and patches to decompose graphs and to track $s$ and $t$ through generating functions. The main result is that the joint distribution of $(s(G),t(G))$ is symmetric, extending a prior bijective result for base cases and providing a refined reduction theory that preserves symmetry across subfamilies of graphs. While a direct bijection remains open, the paper establishes strong structural and enumerative foundations, including confluence of reductions, preservation of induced cycles, and explicit molecular decompositions for key graph classes, which collectively pave the way toward a bijective proof. The work thus contributes a novel, general mechanism for equidistribution on unlabeled graphs and highlights the intricate interplay between graph structure and combinatorial species.

Abstract

We exhibit the joint symmetric distribution of the following two parameters on the set of unlabeled, simple, connected graphs with $n$ vertices. The first parameter is the maximal number of leaves attached to a vertex. The second parameter is the size of the largest set of vertices sharing the same closed neighborhood minus $1$. Apparently, this is the first example of a natural, non-trivial equidistribution of graph parameters on unlabeled connected graphs on a fixed set of vertices. Our proof is enumerative, using the theory of species. Exhibiting an explicit bijection interchanging the two parameters remains an open problem.

Unexpectedly, a symmetry on unlabeled graphs

TL;DR

The paper addresses the problem of finding a natural, nontrivial equidistribution of two graph parameters—tuft number and sibling number —on unlabeled connected graphs with a fixed number of vertices. It develops a comprehensive, species-theoretic framework, introducing co-mating graphs, 3-sort graphs, tufts, and patches to decompose graphs and to track and through generating functions. The main result is that the joint distribution of is symmetric, extending a prior bijective result for base cases and providing a refined reduction theory that preserves symmetry across subfamilies of graphs. While a direct bijection remains open, the paper establishes strong structural and enumerative foundations, including confluence of reductions, preservation of induced cycles, and explicit molecular decompositions for key graph classes, which collectively pave the way toward a bijective proof. The work thus contributes a novel, general mechanism for equidistribution on unlabeled graphs and highlights the intricate interplay between graph structure and combinatorial species.

Abstract

We exhibit the joint symmetric distribution of the following two parameters on the set of unlabeled, simple, connected graphs with vertices. The first parameter is the maximal number of leaves attached to a vertex. The second parameter is the size of the largest set of vertices sharing the same closed neighborhood minus . Apparently, this is the first example of a natural, non-trivial equidistribution of graph parameters on unlabeled connected graphs on a fixed set of vertices. Our proof is enumerative, using the theory of species. Exhibiting an explicit bijection interchanging the two parameters remains an open problem.
Paper Structure (9 sections, 23 theorems, 55 equations, 11 figures)

This paper contains 9 sections, 23 theorems, 55 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Combinatorial Species
  3. A Decomposition of Graphs
  4. Symmetry of the Sibling Number and the Tuft Number
  5. The Reduction of a Graph
  6. Refined Symmetry of the Sibling and the Tuft number
  7. Induced Cycles and Graph Reduction
  8. Co-mating Graphs Reducing to a Given Graph
  9. Towards a Bijection

Key Result

Theorem 1

The sibling number and the tuft number have joint symmetric distribution on the set of unlabeled connected graphs with $n$ vertices. Put differently, the generating series $\sum_{G} x^{s(G)} y^{t(G)}$ is symmetric in $x$ and $y$, where the sum is taken over all unlabeled connected graphs with $n$ ve

Figures (11)

  • Figure 1: Sibling and tuft numbers of graphs on $4$ vertices.
  • Figure 2: The two graphs on $6$ vertices with sibling number $1$ and tuft number $2$, and the two graphs on $6$ vertices with sibling number $2$ and tuft number $1$.
  • Figure 3: The graph on the left is reduced in two steps to the leafless graph without siblings on the right.
  • Figure 4: The first few isomorphism types of the species of connected graphs $\mathcal{G}{}^{\mathrm{c}}$ and the species of $3$-sort graphs $\mathcal{G}_{\!\neq 2}(X, Y, Z)$.
  • Figure 5: A tuft
  • ...and 6 more figures

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Proposition 9
  • Example 10
  • ...and 35 more