Hamiltonian Cycle Polynomial Overview

• Hamiltonian cycle polynomial is an umbrella term covering graph invariants and algebraic complexity constructs, featuring multiple definitions in different research contexts.
• It encodes Hamiltonian cycles by summing monomials over directed cycles or counting k-component 2-factors, thereby connecting combinatorial structures with computational complexity.
• The polynomial facilitates studies on determinant–permanent identities and polynomial-equation encodings, advancing our understanding of NP-hard decision problems.

Searching arXiv for the cited Hamiltonian cycle polynomial papers to ground the article in current arXiv records. {"query":"Hamiltonian cycle polynomial arXiv (Dewan, 25 Jun 2026, Sawczuk et al., 2 Oct 2025, Bo, 2017, 0707.1176)","max_results":10,"sort_by":"relevance"} In current usage, the expression Hamiltonian cycle polynomial does not designate a single universally fixed object. In algebraic complexity, it usually denotes the family $HC_n$, the sum of monomials indexed by directed Hamiltonian cycles in the complete digraph on $n$ labeled vertices. In graph polynomial theory, it also denotes Yi Bo’s invariant $H(G;x)=\sum_{k\ge1} h_k(G)x^k$, where $h_k(G)$ counts $k$-component $2$-factors and $h_1(G)$ is the number of ordinary Hamiltonian cycles. A further strand of work studies determinant–permanent identities for the Hamiltonian-cycle generating function, and another encodes the directed Hamiltonian Cycle Problem as the solvability of a polynomial equation over bounded reals (Dewan, 25 Jun 2026, Bo, 2017, Sawczuk et al., 2 Oct 2025, 0707.1176).

1. Terminology and basic forms

Two definitions dominate the literature, and they arise in different mathematical settings. One is a graph invariant attached to an arbitrary graph $G$. The other is a polynomial family in formal edge variables attached to the complete directed graph on $n$ labeled vertices. Sawczuk and Gnang use the notation $P_{HC_n}(A)$ for the same generating function that algebraic complexity papers denote by $n$0 (Bo, 2017, Dewan, 25 Jun 2026, Sawczuk et al., 2 Oct 2025).

Object	Definition	Role
$n$1	$n$2	Graph invariant from $n$3-component $n$4-factors
$n$5	$n$6	Algebraic-complexity family of weighted Hamiltonian cycles
$n$7	$n$8	Matrix-based notation for the same cycle-generating polynomial
$n$9 encoding of HCP	Single polynomial equation over bounded reals	Decision-problem encoding for directed HCP

This terminological plurality matters because the associated questions differ. For $H(G;x)=\sum_{k\ge1} h_k(G)x^k$0, the focus is deletion–contraction recursion, invariance, and distinguishing power. For $H(G;x)=\sum_{k\ge1} h_k(G)x^k$1, the focus is VNP-completeness, symmetries, equivalence testing, and circuit identities. For determinant–permanent identities, the focus is exact symbolic expansions. For polynomial-equation encodings, the focus is NP-hard feasibility over the reals. This suggests that the phrase Hamiltonian cycle polynomial functions as an umbrella term spanning several adjacent research programs.

2. The graph-invariant polynomial $H(G;x)=\sum_{k\ge1} h_k(G)x^k$2

Yi Bo defines the Hamiltonian cycle polynomial of a graph $H(G;x)=\sum_{k\ge1} h_k(G)x^k$3 by first introducing $H(G;x)=\sum_{k\ge1} h_k(G)x^k$4, the number of $H(G;x)=\sum_{k\ge1} h_k(G)x^k$5-component $H(G;x)=\sum_{k\ge1} h_k(G)x^k$6-factors of $H(G;x)=\sum_{k\ge1} h_k(G)x^k$7. A $H(G;x)=\sum_{k\ge1} h_k(G)x^k$8-component $H(G;x)=\sum_{k\ge1} h_k(G)x^k$9-factor is a spanning subgraph in which every vertex has degree $h_k(G)$0 and which has exactly $h_k(G)$1 connected components, each component being a cycle. The polynomial is then

$h_k(G)$2

In particular, $h_k(G)$3 is the number of ordinary Hamiltonian cycles, so the coefficient of $h_k(G)$4 recovers the usual Hamiltonian-cycle count (Bo, 2017).

The same polynomial is characterized recursively. For any edge $h_k(G)$5 with $h_k(G)$6,

$h_k(G)$7

It is multiplicative on disjoint unions,

$h_k(G)$8

when $h_k(G)$9 and $k$0 have no vertex in common, and its base case on the one-vertex graph with $k$1 loops is

$k$2

The paper proves that this recursion is well-defined, independent of the order in which edges are processed, and that coefficient extraction recovers the numbers $k$3 (Bo, 2017).

The polynomial fits a broader deletion–contraction framework. A general criterion is given for graph-functions satisfying

$k$4

together with multiplicativity and fixed base values. Such a function is well-defined if and only if

$k$5

or else $k$6 is the trivial “vertex-count” invariant. For $k$7, the parameters are $k$8, $k$9, $2$0, which satisfy the criterion. Hence $2$1 is well-defined and invariant under graph isomorphism (Bo, 2017).

The paper also places $2$2 relative to Tutte-type constructions. A seven-variable generalization $2$3 specializes to

$2$4

and also to

$2$5

where $2$6 is the classical Tutte polynomial. In this sense, $2$7 is a one-variable specialization of a two-variable Tutte-type invariant (Bo, 2017).

Several examples calibrate the invariant. A path $2$8 has no spanning $2$9-factor, so $h_1(G)$0. For a cycle $h_1(G)$1 with $h_1(G)$2, the only $h_1(G)$3-factor is the cycle itself, hence $h_1(G)$4. For complete graphs, $h_1(G)$5 and $h_1(G)$6. More generally,

$h_1(G)$7

because $h_1(G)$8 equals the number of ways to partition $h_1(G)$9 into $G$0 cycles, namely the unsigned Stirling numbers of the first kind (Bo, 2017).

Despite encoding the full $G$1-factor spectrum, $G$2 is not a complete invariant. The paper notes that there exist non-isomorphic graphs $G$3 with

$G$4

and that even more refined variants such as the two-variable cycle polynomial $G$5 may still coincide. On the algorithmic side, the defining recursion is analogous to that of the Tutte polynomial. Since even counting ordinary Hamiltonian cycles is $G$6-complete, there is no known polynomial-time algorithm for computing $G$7 on general graphs; the worst-case behavior is exponential-time branching, although certain graph classes admit better methods such as dynamic programming or transfer matrices (Bo, 2017).

3. The algebraic-complexity family $G$8

In algebraic complexity, the Hamiltonian Cycle polynomial is the family

$G$9

where $n$0 is the $n$1 matrix of formal variables and $n$2 is the set of all $n$3-cycles on $n$4. Equivalently, $n$5 may be viewed as the weighted adjacency matrix of the complete directed graph on $n$6 vertices, and $n$7 is the formal sum of the weights of all directed Hamiltonian cycles (Dewan, 25 Jun 2026).

Sawczuk and Gnang use the notation

$n$8

for an $n$9 matrix $P_{HC_n}(A)$0, emphasizing the interpretation as a generating function whose monomials enumerate all directed spanning cycles on $P_{HC_n}(A)$1. When $P_{HC_n}(A)$2 is the adjacency matrix of a directed graph $P_{HC_n}(A)$3, $P_{HC_n}(A)$4 is exactly the number of Hamiltonian cycles in $P_{HC_n}(A)$5 (Sawczuk et al., 2 Oct 2025).

The complexity-theoretic status of the family is classical. Valiant showed that the permanent family is VNP-complete over every field of characteristic other than $P_{HC_n}(A)$6, and that $P_{HC_n}(A)$7 is VNP-complete over every field, and even every commutative ring. The 2026 paper therefore describes $P_{HC_n}(A)$8 as the “universal” VNP-complete graph-polynomial, in contrast to the permanent family, whose completeness fails in characteristic $P_{HC_n}(A)$9 (Dewan, 25 Jun 2026).

Subsequent work cited there situates $n$00 within several major lines of algebraic complexity. Jerrum–Snir gave exact monotone-circuit lower bounds for $n$01. Hüttenhain–Ikenmeyer gave exact binary-determinantal-complexity bounds. Goulden–Jackson and Grochow related $n$02 to immanants. Malod, Grochow–Mulmuley–Qiao, and Hrubes used the VNP-completeness of $n$03 to obtain results uniformly across fields and, in the case of Hrubes, to exhibit new VNP-complete families in characteristic $n$04 (Dewan, 25 Jun 2026).

For $n$05, the symmetry analysis treats $n$06 as a homogeneous degree-$n$07 polynomial in the $n$08 variables $n$09 with $n$10. This presentation removes diagonal variables from the ambient coordinate space while preserving the Hamiltonian-cycle monomial structure (Dewan, 25 Jun 2026).

4. Symmetries, equivalence testing, and circuit identities

A central structural question is how much of $n$11 is captured by its symmetry group. For a polynomial $n$12, the symmetry group is

$n$13

and the Lie algebra is

$n$14

For $n$15, a matrix $n$16 lies in $n$17 if and only if it is diagonal and its diagonal entries satisfy the cycle-sum constraints

$n$18

for every $n$19-cycle $n$20. The dimension of $n$21 is $n$22 for $n$23; for $n$24, the dimension is $n$25 if $n$26 and $n$27 if $n$28 (Dewan, 25 Jun 2026).

The symmetry group itself has a concrete description over sufficiently large fields. If

$n$29

then every symmetry $n$30 factors uniquely as

$n$31

where $n$32 is a permutation-matrix symmetry, acting by relabeling $n$33 or by transpose, and $n$34 is a diagonal scaling matrix satisfying the same cycle-sum constraints. This parallels the permanent in one respect, but not in all respects: for $n$35 and sufficiently large fields, $n$36 is not determined up to scale by its symmetry group. The paper states that there exists $n$37 with

$n$38

That non-characterization distinguishes $n$39 from the permanent polynomial in the sense used there (Dewan, 25 Jun 2026).

The same paper studies equivalence testing. Given black-box access to a polynomial $n$40, the problem is to decide whether $n$41 is equivalent to $n$42 under an invertible linear change of variables. Its main theorem states that if

$n$43

then there is a randomized poly$n$44-time algorithm that decides with high probability whether there exists $n$45 such that

$n$46

and outputs $n$47 when such an equivalence holds. The algorithm proceeds by a Lie-algebraic reduction to permutation-and-scaling form, recovery of the permutation from the zero-pattern of mixed second derivatives, recovery of the scaling by solving a linear system over cyclic monomials, and a final verification stage (Dewan, 25 Jun 2026).

The verification stage uses downward self-reducibility rather than symmetry characterization. For every $n$48, there is an explicit $n$49 matrix $n$50, constructible in $n$51 time, such that

$n$52

Moreover, $n$53 is characterized by the $n$54 identities

$n$55

together with

$n$56

From this the paper derives a circuit-identity characterization, a randomized poly$n$57-time algorithm for testing whether a circuit $n$58 computes $n$59, and a Flip theorem for $n$60: assuming $n$61 has no poly$n$62-size circuits over fields with $n$63, there is a randomized poly$n$64 algorithm outputting matrices $n$65 such that every size-$n$66 circuit fails on at least one of them (Dewan, 25 Jun 2026).

5. Determinant–permanent identities

A separate line of work studies exact symbolic identities for the Hamiltonian-cycle generating polynomial. For an $n$67 matrix $n$68, Sawczuk and Gnang prove the Hamiltonian-cycle identity

$n$69

with the conventions $n$70 and $n$71. Here $n$72 is the principal submatrix of $n$73 indexed by $n$74, and $n$75 is the sum of monomials corresponding to single $n$76-cycles (Sawczuk et al., 2 Oct 2025).

The first proof is combinatorial. Expanding

$n$77

produces monomials corresponding to cycle covers of $n$78. Fixing a permutation $n$79 with cycle decomposition

$n$80

the monomial $n$81 appears in $n$82 exactly when each cycle of $n$83 lies entirely in $n$84 or entirely in $n$85. Summing the resulting coefficients over all $n$86 yields

$n$87

so only the single-cycle permutations survive (Sawczuk et al., 2 Oct 2025).

The second proof is symbolic and uses Tutte’s Directed Matrix-Tree Theorem, a determinant-sum lemma, and a multivariable product rule. The construction starts from a polynomial listing of rooted directed spanning trees, passes to unicyclic functional graphs by replacing $n$88 with $n$89, and then applies the full partial-derivative operator

$n$90

to isolate exactly those unicycles in which every vertex has in-degree $n$91, namely Hamiltonian cycles. After expanding the determinant via subsets and applying the product rule, the surviving terms are precisely the determinant–permanent summands above (Sawczuk et al., 2 Oct 2025).

The same paper derives an analogous Hamiltonian-path identity. Defining a Hamiltonian-path polynomial $n$92 as the sum over directed spanning paths of length $n$93, it proves

$n$94

The cycle and path identities express $n$95-type generating functions in terms of the permanent and determinant. The paper explicitly notes that these identities do not yield an efficient algorithm in general, since the permanent remains a $n$96-complete object (Sawczuk et al., 2 Oct 2025).

6. Polynomial-equation encodings of the Hamiltonian cycle problem

Another nearby use of polynomial language is not a graph polynomial invariant but an exact encoding of the decision version of the directed Hamiltonian Cycle Problem. Chermakani demonstrates a polynomial approach that expresses directed HCP as the solvability of a single polynomial equation within a bounded real space. The construction begins with four new theorems about periodic functions with irrational periods, including uniqueness results for tuples of next-high gaps and infinitely many simultaneous overlaps for collections of such functions (0707.1176).

The encoding then uses a trigonometric substitution for the $n$97 directed edges: $n$98 where $n$99 records whether the directed edge $H(G;x)=\sum_{k\ge1} h_k(G)x^k$00 is present. By expanding $H(G;x)=\sum_{k\ge1} h_k(G)x^k$01 through

$H(G;x)=\sum_{k\ge1} h_k(G)x^k$02

and then recursively expressing $H(G;x)=\sum_{k\ge1} h_k(G)x^k$03 and $H(G;x)=\sum_{k\ge1} h_k(G)x^k$04 in terms of $H(G;x)=\sum_{k\ge1} h_k(G)x^k$05 and $H(G;x)=\sum_{k\ge1} h_k(G)x^k$06, each $H(G;x)=\sum_{k\ge1} h_k(G)x^k$07 becomes a polynomial in the four master variables

$H(G;x)=\sum_{k\ge1} h_k(G)x^k$08

times the known constant $H(G;x)=\sum_{k\ge1} h_k(G)x^k$09 (0707.1176).

Five sets of constraints are imposed. They force the $H(G;x)=\sum_{k\ge1} h_k(G)x^k$10 to behave like binary edge-selection variables, require exactly one outgoing chosen edge at each vertex,

$H(G;x)=\sum_{k\ge1} h_k(G)x^k$11

enforce that the chosen edges form a single cycle of length $H(G;x)=\sum_{k\ge1} h_k(G)x^k$12, impose the trigonometric identities

$H(G;x)=\sum_{k\ge1} h_k(G)x^k$13

and bound approximation errors by $H(G;x)=\sum_{k\ge1} h_k(G)x^k$14. Auxiliary signal variables $H(G;x)=\sum_{k\ge1} h_k(G)x^k$15 propagate a unit signal through the selected edges so that the final-time conditions exclude smaller disjoint cycles. After converting inequalities to equations with slack variables, all constraints are merged into one polynomial

$H(G;x)=\sum_{k\ge1} h_k(G)x^k$16

By construction, $H(G;x)=\sum_{k\ge1} h_k(G)x^k$17 has a real solution if and only if the graph has a directed Hamiltonian cycle (0707.1176).

All variables lie in fixed bounded intervals: the trigonometric variables lie in $H(G;x)=\sum_{k\ge1} h_k(G)x^k$18, the error variables lie in $H(G;x)=\sum_{k\ge1} h_k(G)x^k$19, and the signal and slack variables are bounded as well. The paper states that any directed HCP instance can be translated in polynomial time into such a single polynomial-equation feasibility problem, and concludes that feasibility of a single polynomial equation over bounded real intervals is NP-hard. It further points to a prospective consequence: if one could decide in polynomial time whether a constant-variable polynomial equation has a real solution in a bounded box, then one would solve NP-complete problems by that subroutine, implying $H(G;x)=\sum_{k\ge1} h_k(G)x^k$20 (0707.1176).

This construction is distinct from both $H(G;x)=\sum_{k\ge1} h_k(G)x^k$21 and $H(G;x)=\sum_{k\ge1} h_k(G)x^k$22, but it belongs to the same broader landscape in which Hamiltonian-cycle structure is recast into polynomial form. A plausible implication is that the Hamiltonian cycle polynomial serves not only as an invariant or generating function, but also as a bridge between combinatorial graph structure, algebraic identities, and decision-problem encodings.