Graphical Bell Numbers in Graph Theory

• Graphical Bell Numbers are defined as the total count of proper partitions of a graph's vertex set, generalizing classical Bell numbers by considering interchangeable color classes.
• They are computed via partition and chromatic polynomials, with explicit formulas developed for families such as complete multipartite graphs, bipartite graphs minus a matching, and Mycielskians.
• The study of Graphical Bell Numbers provides insights into hereditary graph properties, asymptotic inequalities, and enumerative combinatorial techniques with applications in quantum normal-ordering and pattern avoidance.

A graphical Bell number, denoted $B(G)$ for a graph $G = (V,E)$, generalizes the classical Bell number by counting the total number of proper partitions of the vertex set—partitions into nonempty blocks such that no block contains an edge of %%%%2%%%%. Equivalently, these correspond to proper vertex-colorings of $G$ with interchangeable colors, i.e., color-classes where the names of colors are irrelevant, thus measuring the number of stable partitions or non-equivalent colorings. This construction naturally bridges set partition combinatorics, graph coloring theory, chromatic polynomials, and hereditary property growth thresholds (Allagan et al., 7 Dec 2025, Hertz et al., 2021, Codara et al., 2013, Atminas et al., 2014).

1. Fundamental Definitions and Properties

For a simple graph $G$ of order $n = |V|$:

• $S(G;k)$: Number of proper partitions of $V$ into exactly $k$ blocks, i.e., colorings using $k$ interchangeable colors without edges in any block.
• $B(G) = \sum_{k=1}^n S(G;k)$: Total number of proper partitions, the graphical Bell number.

These can be encoded via the partition polynomial $F(G;x) = \sum_{k=1}^n S(G;k)x^k$, and relate to the chromatic polynomial $\chi(G;x) = \sum_{k=1}^n S(G;k)x^{\underline{k}}$ where $x^{\underline{k}} = x(x-1)\cdots(x-k+1)$ (Allagan et al., 7 Dec 2025).

The Burnside orbit-counting method provides a closed-form for $B(G)$:

$$B(G) = \sum_{k=0}^n \frac{1}{k!} \sum_{\pi \in S_k} \chi_G(m_1(\pi))$$

where $m_1(\pi)$ is the number of fixed points under permutation $\pi$ and $\chi_G(x)$ is the chromatic polynomial (Hertz et al., 2021).

Special cases:

• $G = E_n$ (empty graph): $S(E_n;k) = S(n,k)$, $B(E_n) = B_n$ (classical Bell number).
• $G = K_n$ (complete graph): $B(K_n) = 1$.
• Tree $T$ of $n$ vertices: $S(T;k) = S(n-1,k)$, $B(T) = B_{n-1}$ (Hertz et al., 2021).

2. Explicit Formulas for Graph Families

Complete Multipartite Graphs

For $G = K(n_1, \ldots, n_\ell)$:

$$S(G;k) = \sum_{j_1+\cdots+j_\ell = k,\,j_i \geq 1} \prod_{i=1}^\ell S(n_i, j_i)$$

and

$B(K(n_1, ..., n_\ell)) = \prod_{i=1}^\ell B_{n_i}$

For $K_{n,n}$: $B(K_{n,n}) = B_n^2$; for $K_{n,n,n}$: $B_n^3$ (Allagan et al., 7 Dec 2025).

Bipartite Graph Minus Matching

For $H = K_{n,n} - M$:

$B(K_{n,n} - M) = \sum_{k=0}^n \binom{n}{k} B_k^2$

This interpolation formula reflects the structure moving between $K_{n,n}$ and $E_n \cup E_n$ by edge deletion (Allagan et al., 7 Dec 2025).

Mycielskians of Trees

For the Mycielskian of the star $M(St_n)$:

• $$B(M(St_n)) = 2 \sum_{k=0}^{m} \binom{m}{k} B_{2m-k} + \sum_{i=0}^m \sum_{j=0}^m \binom{m}{i} \binom{m}{j} B_{2m-i-j}$$ where $m = n-1$.
• For Stirling numbers:
  • $S(M(St_n);3) = 2^n + 1$
  • $S(M(St_n);2n) = 2n^2 - 3n + 3$

The structure generalizes to arbitrary trees via subset summation over leaf choices and induced classical Bell numbers (Allagan et al., 7 Dec 2025).

3. Graphical Bell Numbers and Generalized Bell Numbers

The concept relates to generalized Bell numbers $B_{r,s}(n)$ defined via combinatorial models such as graph colorings and labeled Eulerian digraphs. In the case $r = s = m$,

• $$B_{m,m}(n,t) = \sum_{k=m}^{mn} S_{m,m}(n,k) t^k = e^{-t} \sum_{k=0}^\infty \frac{(k)_m^n}{k!} t^k$$
• $$B_{m,m}(n) = \sum_{k=m}^{mn} S_{m,m}(n,k) = e^{-1} \sum_{j=0}^\infty \frac{(j)_m^n}{j!}$$

Interpretation:

• $S_{m,m}(n,k)$ counts the number of $k$-colorings of $nK_m$ (disjoint union of $n$ cliques of size $m$) into stable sets (Codara et al., 2013).
• These numbers coincide with the count of $(n,m)$-labelled Eulerian digraphs with $k$ vertices.

For $m=1$, the classical Bell numbers are recovered; for arbitrary $m$, the combinatorial and recurrence structure is preserved (Codara et al., 2013).

4. Hereditary Properties and the Bell Threshold

Hereditary graph property speeds exhibit a sharp threshold at the classical Bell number $B_n$:

• A hereditary property $P$ has speed below the Bell number if $P_n = o(B_n)$, or above if $P_n \geq B_n$ for all large $n$.
• The threshold is characterized by the so-called "distinguishing number" $k_X$:
  • If $k_X = \infty$, $P$ contains one of 13 minimal classes (e.g., disjoint unions of cliques, star forests, threshold graphs).
  • If $k_X < \infty$, minimal classes above $B_n$ form an infinite family, parametrized by periodic words and finite graphs $H$.

A concrete algorithm decides whether a given finitely forbidden set $F$ defines a hereditary class above or below $B_n$:

1. Test for containment of one of the 13 traditional minimal classes (checking $k_X = \infty$).
2. Otherwise, examine periodic constructions via strips and cyclic words to certify above/below status (Atminas et al., 2014).

5. Inequalities, Asymptotics, and Enumerative Properties

The average-color methodology links graphical Bell numbers to tight inequalities for classical Bell numbers:

• Strict log-convexity: $B_n^2 < B_{n-1} B_{n+1}$
• Sum-bound inequality: $B_n(B_n + B_{n+1}) < B_{n-1}(B_{n+1} + B_{n+2})$

These are derived by comparing average color counts across graph operations and partitions. Burnside's lemma and partition polynomial evaluations yield further enumerative results for paths, cycles, stars, and multipartite families (Hertz et al., 2021). Asymptotically, the Bell numbers satisfy $\ln B_n / n = \ln n - \ln \ln n + \Theta(1)$ (Atminas et al., 2014).

6. Connections to Integer Sequences and Pattern Avoidance

Explicit counts for graphical Bell numbers correspond to known OEIS integer sequences: | Graphical Structure | OEIS Entry | Formula/Interpretation | |--------------------------------------|--------------|-----------------------------------------------------------------------------| | $S(M(St_n);3)$ | A000051 | $2^n+1$ (proper $3$-partitions of Mycielskian star) | | $S(M(St_{n+2});2(n+2))$ | A096376 | $2n^2 + n + 2$ (up to index-shift) | | $B(K_{n,n};4)$ | A384980 | $2 - 2^{n+1} + 3^{n-1} + 4^{n-1}$ (proper $4$-partitions of $K_{n,n}$) | | $B(K_{n,n};5)$ | A384981 | $$6^{n-1} - (5/3)2^{2n-2} - 2 \cdot 3^{n-1} + 2^{n+1} - 4/3$$ | | $B(K_{n,n,n};5)$ | A384988 | $$\frac{1}{4}(18-18 \cdot 2^n + 2 \cdot 3^n + 3 \cdot 4^n)$$ | | Triangular arrays for $K_{n,n,n}$ | A385432 | $T(n,k)=B(K_{n,n,n};k)$, $3 \leq k \leq 3n$ | | Triangular arrays for $K_{n,n}-M$ | A385437 | $T(n,k)=B(K_{n,n}-M;k)$, $2 \leq k \leq 2n$ |

Patterns in color-avoidance, partition rigidity, and block size constraints lead to links with classical and modern combinatorics (Allagan et al., 7 Dec 2025).

7. Applications and Open Directions

Graphical Bell numbers appear in quantum normal-ordering combinatorics, the enumerative combinatorics of graph colorings, decision algorithms for property speeds, and inequalities governing growth and structure of integer sequences. Explicit formulas for multipartite and edge-deleted graphs provide computationally tractable invariants. Open problems include the polynomial-time decidability of the threshold phenomenon, precise bounds for algorithmic steps, and broader generalizations to ordered Bell numbers and related jump sequences in hereditary property speeds (Atminas et al., 2014).