Path-Cover Polynomials in Graph Theory

• Path-Cover Polynomials are graph polynomials that enumerate vertex-disjoint path covers, linking to structures like matching and clique-cover polynomials.
• They utilize analytic representations via associated Laguerre polynomials and duality relations to connect with other key graph invariants.
• Efficient algorithms on cographs and multipartite graphs demonstrate their practical impact in counting Hamiltonian paths and cycles.

A path-cover polynomial is a graph polynomial that encodes, for a given finite graph, detailed information about the enumeration of vertex-disjoint path covers—partitionings of the vertex set into collections of vertex-disjoint paths, each covering part or all of the graph. Path-cover polynomials sit at the intersection of algebraic and enumerative graph theory, closely related to matching, clique-cover, and chromatic polynomials. Recent work has established duality theorems, analytic representations in terms of the associated Laguerre polynomials, and provided efficient algorithms for their computation in certain graph families (Jumadildayev, 17 Dec 2025), as well as explicit combinatorial and generating function expansions through the framework of universal edge elimination polynomials (Dohmen, 2014).

1. Definition and Basic Properties

Let $G=(V,E)$ be a simple graph or digraph of order $n=|V|$. A $k$-path cover is a collection of $k$ vertex-disjoint directed paths whose union of vertices equals $V$. Trivial paths (length $0$, single-vertex paths) are allowed. Let $p_k(G)$ be the number of $k$-path covers of $G$. The path-cover polynomial is defined by

$\pi_G(t) = \sum_{k=1}^n p_k(G)\,t^k,$

with a signed variant

$\pi_G^+(t) = \sum_{k=1}^n(-1)^{n-k}\,p_k(G)\,t^k,$

such that $\pi_G(t) = (-1)^n \pi_G^+(-t)$.

For the classical case on simple (undirected) paths $P_n$, the polynomial reduces to

$$\Pi(P_n;z) = \sum_{k=1}^n \binom{n-1}{k-1} z^k = z(1+z)^{n-1}.$$

Each $p_k(G)$ counts the number of ways to partition $V$ into $k$ vertex-disjoint paths; $p_n(G)=1$ (all trivial paths), $p_1(G)$ is the number of directed Hamiltonian paths in $G$ (Jumadildayev, 17 Dec 2025, Dohmen, 2014).

2. Combinatorial Interpretation and Specializations

The coefficients $p_k(G)$ enumerate decompositions of $G$ into $k$ disjoint paths, generalizing matching and Hamiltonian path/cycle enumerators. For the path $P_n$, the coefficient $p_k(P_n) = \binom{n-1}{k-1}$ counts ways to partition the $n$-vertex path into $k$ contiguous intervals, matching the classical combinatorial interpretation of path covers.

The path-cover polynomial specializes from the two-variable covered-components polynomial $C(G;u,v)$ (Dohmen, 2014), with

$\Pi(G;z) = C(G;1,z+1),$

enabling explicit formulae through closed-form expressions for $C(G;u,v)$. For $P_n$,

$$\Pi(P_n;z) = \sum_{k=1}^n \binom{n-1}{k-1} z^k = z(1+z)^{n-1}.$$

The corresponding generating function is

$$\sum_{n=1}^\infty \Pi(P_n;z) t^n = \frac{z t}{1-(1+z)t}.$$

3. Duality Theorem and Structural Relations

Path-cover polynomials satisfy a duality relation paralleling those for matching and clique-cover polynomials. For a simple undirected graph $G$ with complement $\overline{G}$, and setting $D=\frac{d}{dt}$, the differential operator $\phi_{\pi} = \exp(t D^2)$ yields the central duality: $$\pi_G(t) = \phi_{\pi}\left[ \pi_{\overline G}^+(t) \right].$$ This is explicitly realized by an inclusion–exclusion sum over the number of $k$-path covers: $$p_k(G) = \sum_{j=0}^{n-1} (-1)^j\, p_k(K_{n-j})\, p_{n-j}(\overline G),$$ where $p_k(K_m)$ counts the $k$-path covers of the $m$-clique.

This duality provides analytic links to other graph polynomials (matching, clique-cover, chromatic), all of which satisfy similar relations mediated via appropriate differential operators (Jumadildayev, 17 Dec 2025).

4. Analytic Representations and Orthogonality

A Lebesgue–Stieltjes integral associated with the measure $d\mu(t)=t^{-1}e^{-t}dt$ enables analytic expressions for path-cover polynomials: $$\mathcal L[f] = \int_0^\infty f(t)\, d\mu(t) = \int_0^\infty t^{-1}e^{-t} f(t)\, dt.$$ The induced inner product,

$$\langle f, g \rangle = \int_0^\infty t^{-1}e^{-t} f(t) g(t) dt,$$

connects path-cover polynomials to the system of associated Laguerre polynomials $L_n^{(-1)}(t)$: $$L_n^{(-1)}(t) = \frac{(-1)^n}{n!} \exp(-t D^2)[t^n], \qquad (-1)^n n! L_n^{(-1)}(t) = \pi_{K_n}^+(t),$$ establishing that $\{L_n^{(-1)}(t)\}$ forms an orthogonal family for this inner product.

For any graph $G$ with $n$ vertices and $c(G)$ the number of Hamiltonian cycles,

$$\mathcal{L}\left[\pi_{\overline G}^+(t)\right] = c(G) + (-1)^{n-1} c(\overline G).$$

If $\overline G$ has no Hamiltonian cycles, then $\mathcal{L}[\pi_{\overline G}^+] = c(G)$ (Jumadildayev, 17 Dec 2025).

5. Explicit Enumeration in Multipartite Graphs

The join-factorization theorem enables closed-form enumeration of Hamiltonian paths and cycles in complete multipartite graphs $K_{\alpha_1,\ldots,\alpha_m}$: $$\pi_{K_{\alpha_1,\ldots,\alpha_m}}(t) = \phi_\pi \left[ \prod_{i=1}^m \phi_\pi^{-1}[t^{\alpha_i}] \right],$$ allowing the number of directed Hamiltonian paths (coefficient of $t^1$) to be extracted.

Alternatively, the count appears as a mixed moment of associated Laguerre polynomials under the $\mathrm{Exp}(1)$ distribution: $$\#\{\text{Hamiltonian paths}\} = \mathbb{E}\left[ \prod_{i=1}^m (-1)^{\alpha_i} \alpha_i! L_{\alpha_i}^{(-1)}(X) \right], \quad X \sim \mathrm{Exp}(1),$$ and may also be reformulated via partial Bell polynomials in the balanced case. For Hamiltonian cycles in complete $(m+1)$-partite graphs, a similar explicit formula holds, reducible in the regular case to an $O((nm)\log(nm)\log m)$ summation (Jumadildayev, 17 Dec 2025).

6. Algorithmic Evaluation on Cographs

A cograph is any graph built from single vertices via repeated disjoint-union ($\cup$) and join ($\triangledown$) operations. Every cograph admits a unique cotree $T_G$, with each internal node labeled by union or join. For any graph polynomial $P$ in this family (matching, path-cover, clique-cover, chromatic), the following decompositions hold: $$P_{G\triangledown H} = \phi[P_G^{(-1)} P_H^{(-1)}], \quad P_{G\cup H} = P_G \cdot P_H,$$ allowing $P_G$ to be computed by a bottom-up traversal of $T_G$.

By precomputing $\phi[t^k]$ and $\phi^{-1}[t^k]$ for $k \leq n$ and using FFT-based convolution, each step costs $O(n \log n)$; the cotree contains $O(n)$ nodes for an $n$-vertex cograph, yielding a total complexity of $O(n^2 \log n)$. This approach efficiently computes $\pi_G(t)$ for any cograph, allowing direct extraction of the number of Hamiltonian paths and cycles (Jumadildayev, 17 Dec 2025).

7. Connections to Universal Edge Elimination and Generating Functions

Path-cover polynomials arise as a specialization of the universal edge elimination polynomial $\xi(G;x,y,z)$, defined recursively via edge deletion, contraction, and edge-terminal elimination. For the path $P_n$,

$\Pi(P_n;z) = \sum_{k=1}^n \binom{n-1}{k-1} z^k,$

and the generating function is rational: $$F_{P}(t) = \sum_{n=0}^\infty \xi(P_n;x,y,z) t^n = \frac{1-yt}{1 - (x+y)t - z t^2}.$$ Specializations recover classical expansions and binomial sum expressions for path-cover enumerations, offering another analytic route for enumeration in paths and related graph classes (Dohmen, 2014).

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The synthesis of duality, analytic, and combinatorial frameworks for path-cover polynomials provides a foundational toolkit both for theoretical analysis and algorithmic computation in the study of path decompositions in graphs.