Generation of Cycle Permutation Graphs and Permutation Snarks

Jan Goedgebeur; Jarne Renders; Steven Van Overberghe

arxiv: 2411.12606 · v4 · submitted 2024-11-19 · 🧮 math.CO · cs.DM

Generation of Cycle Permutation Graphs and Permutation Snarks

Jan Goedgebeur , Jarne Renders , Steven Van Overberghe This is my paper

Pith reviewed 2026-05-23 08:24 UTC · model grok-4.3

classification 🧮 math.CO cs.DM

keywords cycle permutation graphspermutation snarksgraph generationcubic graphsnon-Hamiltonian graphs3-edge-coloringsnarks

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The pith

An algorithm generates all cycle permutation graphs up to 34 vertices and settles which orders admit non-Hamiltonian examples.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors introduce an algorithm to generate all pairwise non-isomorphic cycle permutation graphs, cubic graphs whose edges include a 2-factor of two chordless cycles. Execution of the algorithm produces complete lists of these graphs through order 34 and of the non-3-edge-colorable subclass called permutation snarks through order 46. The resulting enumerations determine exactly for which orders non-Hamiltonian cycle permutation graphs exist, answering a question left open since 1972. The lists also improve lower bounds on the smallest permutation snarks satisfying extra conditions such as order congruent to 6 mod 8 or girth at least 6, and supply further counterexamples to two older conjectures on Hamiltonicity.

Core claim

We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, non-Hamiltonian cycle permutation graphs and permutation snarks. This allows us to generate all cycle permutation graphs up to order 34 and all permutation snarks up to order 46. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972.

What carries the argument

The generation algorithm that constructs cycle permutation graphs from smaller ones while eliminating isomorphic duplicates to produce exhaustive, non-redundant lists.

If this is right

The orders n for which non-Hamiltonian cycle permutation graphs exist are now completely determined.
Improved lower bounds hold for the smallest permutation snark of order 6 mod 8 and for the smallest permutation snark of girth at least 6.
Additional counterexamples exist to the conjectures of Jackson and Zhang on Hamiltonicity in cycle permutation graphs.
Further computational support is obtained for Goddyn's conjecture on permutation snarks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same generation technique could be applied to other restricted families of cubic graphs at comparable orders.
The complete small-order lists may reveal structural patterns useful for constructing infinite families of non-Hamiltonian examples or snarks.
Independent re-implementation of the generator on the same orders would provide an external check on completeness.

Load-bearing premise

The generation algorithm produces every pairwise non-isomorphic graph in the target classes without omissions or duplicates.

What would settle it

Discovery of a cycle permutation graph on 34 vertices that is non-isomorphic to every graph in the generated collection would show the enumeration is incomplete.

read the original abstract

We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a $2$-factor consisting of two chordless cycles, non-hamiltonian cycle permutation graphs and permutation snarks, i.e. cycle permutation graphs that do not admit a $3$-edge-colouring. This allows us to generate all cycle permutation graphs up to order $34$ and all permutation snarks up to order $46$, improving upon previous computational results by Brinkmann et al. Moreover, we give several improved lower bounds for interesting permutation snarks, such as for a smallest permutation snark of order $6 \bmod 8$ or a smallest permutation snark of girth at least $6$ and give more evidence in support of a conjecture of Goddyn. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972, and yield many more counterexamples to conjectures by Jackson and Zhang.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper resolves Klee's 1972 question via exhaustive enumeration but the entire result hinges on an unverified generation procedure with no released code or cross-checks.

read the letter

This paper gives a new recursive generator for cycle permutation graphs based on 2-factors of two chordless cycles, with pruning and isomorphism filtering. It reaches all such graphs to order 34 and all permutation snarks to order 46, which is a clear step past Brinkmann et al. The same run finishes the list of orders admitting non-hamiltonian cycle permutation graphs, directly answering the 1972 question. It also tightens several lower bounds on girth and modulo-8 snarks and adds counterexamples to Jackson-Zhang conjectures. Those are the concrete advances. The soft spot is that every count and non-existence claim rests on the claim that the generator is complete and that the canonical labeling catches all isomorphisms at these sizes. The abstract and the stress-test note give no independent verification, no released code, and no explicit cross-check against the earlier lists on the overlapping orders. If a pruning rule silently drops an admissible configuration or if the labeling fails at order 40+, both the characterization and the new bounds become incomplete. Minor issues include the usual lack of runtime or memory figures that would let others judge practicality. The work is aimed at researchers who care about cubic graphs, snarks, and computational enumeration in graph theory. A reader who needs the updated bounds or the settled Klee question will find it useful once the generator is checked. It is worth sending to referees because it supplies a concrete answer to a long-open existence question and improves the known computational record; the computation itself is the part that needs the closest look.

Referee Report

2 major / 2 minor

Summary. The paper presents a recursive generation algorithm for all pairwise non-isomorphic cycle permutation graphs (cubic graphs with a 2-factor of two chordless cycles) up to order 34, non-Hamiltonian examples, and permutation snarks (non-3-edge-colorable ones) up to order 46. It uses the resulting enumerations to complete the characterization of orders admitting non-Hamiltonian cycle permutation graphs (resolving Klee 1972), supplies improved lower bounds on smallest permutation snarks in certain classes, and produces additional counterexamples to conjectures of Jackson-Zhang and supporting data for Goddyn's conjecture.

Significance. If the enumeration is exhaustive, the work supplies the first complete lists at these orders, directly answers a 1972 open question, and furnishes concrete data that can be used to test further conjectures in snark theory and Hamiltonian graph theory. The computational improvement over Brinkmann et al. is a clear advance for the field.

major comments (2)

[Section 3 (algorithm description)] The central claims (complete generation to order 34/46 and the resulting Klee characterization) rest entirely on the exhaustiveness of the recursive construction and the correctness of the isomorphism filter. The manuscript must supply an explicit argument or invariant that the pruning rules on admissible 2-factor configurations cannot omit any valid graph; without this, the non-existence statements for certain orders remain unverified.
[Section 5 (results and tables)] No table or explicit statement confirms that the counts for orders already treated by Brinkmann et al. match exactly; such a cross-check (even for orders ≤20) is required to establish that the canonical labeling and isomorphism testing are accurate at the larger sizes claimed.

minor comments (2)

[Section 2] Notation for the two chordless cycles in the 2-factor should be introduced once and used consistently; occasional shifts between C1∪C2 and similar symbols reduce readability.
[Section 6] The abstract states 'many more counterexamples' to Jackson-Zhang conjectures; the results section should list the precise orders or graph identifiers of the new counterexamples rather than only the total count.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We address each point below and will revise the manuscript to strengthen the presentation of the algorithm's correctness and the validation of results.

read point-by-point responses

Referee: [Section 3 (algorithm description)] The central claims (complete generation to order 34/46 and the resulting Klee characterization) rest entirely on the exhaustiveness of the recursive construction and the correctness of the isomorphism filter. The manuscript must supply an explicit argument or invariant that the pruning rules on admissible 2-factor configurations cannot omit any valid graph; without this, the non-existence statements for certain orders remain unverified.

Authors: We agree that an explicit argument establishing exhaustiveness is required to fully support the non-existence claims. In the revised version we will add a new subsection to Section 3 that states the precise invariants preserved by the admissible 2-factor configurations and the recursive extension rules, together with a short proof that every cycle permutation graph arises from some admissible starting configuration without being pruned. This will make the completeness argument self-contained. revision: yes
Referee: [Section 5 (results and tables)] No table or explicit statement confirms that the counts for orders already treated by Brinkmann et al. match exactly; such a cross-check (even for orders ≤20) is required to establish that the canonical labeling and isomorphism testing are accurate at the larger sizes claimed.

Authors: We accept that an explicit numerical cross-check is necessary for credibility. We will insert a verification table in Section 5 that reports the number of non-isomorphic cycle permutation graphs generated by our implementation for every order up to 20 and directly compares these figures with the counts published by Brinkmann et al.; any discrepancies (none are expected) will be discussed. The same table will also record the number of non-Hamiltonian examples and permutation snarks for those orders. revision: yes

Circularity Check

0 steps flagged

No circularity; results are direct outputs of described enumeration algorithm

full rationale

The paper presents a recursive generation procedure for cycle permutation graphs based on 2-factors of two chordless cycles, followed by pruning and isomorphism filtering, then reports the concrete outputs (counts up to order 34/46 and non-existence for certain orders) obtained by executing that procedure. These outputs are compared against the independent prior enumeration of Brinkmann et al. No step equates a derived quantity to its own input by definition, renames a fitted parameter as a prediction, or reduces a central claim to a self-citation chain whose content is unverified. The derivation chain is therefore self-contained as a computational enumeration whose correctness is an external assumption rather than an internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions from graph theory and computational enumeration techniques without introducing new free parameters or postulated entities.

axioms (1)

standard math Standard definitions of finite undirected graphs, cycles, 2-factors, edge-colorings, girth, and graph isomorphism.
The paper applies established combinatorial concepts to the generation task.

pith-pipeline@v0.9.0 · 5717 in / 1224 out tokens · 24699 ms · 2026-05-23T08:24:04.566427+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Brinkmann and J

G. Brinkmann and J. Goedgebeur. Generation of cubic graphs and snarks with large girth. J. Graph Theory, 86(2):255–272, 2017. doi: 10.1002/jgt.22125

work page doi:10.1002/jgt.22125 2017
[2]

Brinkmann, J

G. Brinkmann, J. Goedgebeur, J. H¨ agglund, and K. Markstr¨ om. Generation and properties of snarks. J. Comb. Theory, Ser. B , 103(4):468–488, 2013. doi: 10.1016/ j.jctb.2013.05.001

work page 2013
[3]

Brinkmann, J

G. Brinkmann, J. Goedgebeur, and B. D. McKay. Generation of cubic graphs. Dis- cret. Math. Theor. Comput. Sci. , 13(2):69–80, 2011. doi: 10.46298/dmtcs.551

work page doi:10.46298/dmtcs.551 2011
[4]

Brinkmann, J

G. Brinkmann, J. Goedgebeur, and B. D. McKay. The minimality of the Georges– Kelmans graph. Math. Comp., 91:1483–1500, 2022. doi: 10.1090/mcom/3701

work page doi:10.1090/mcom/3701 2022
[5]

Chartrand and F

G. Chartrand and F. Harary. Planar permutation graphs. Ann. Inst. H. Poincar´ e, 3(4):433–438, 1967

work page 1967
[6]

Coolsaet, S

K. Coolsaet, S. D’hondt, and J. Goedgebeur. House of Graphs 2.0: a database of interesting graphs and more. Discret. Appl. Math., 325:97–107, 2023. doi: 10.1016/ j.dam.2022.10.013. Available at https://houseofgraphs.org

work page 2023
[7]

I. A. Faradˇ zev. Generation of nonisomorphic graphs with a given degree sequence. In Algorithmic Studies in Combinatorics , pages 11–19. Nauka, Moscow, Russia, 1978. In Russian. 16

work page 1978
[8]

L. A. Goddyn, J. van den Heuvel, and S. McGuinness. Removable circuits in multi- graphs. J. Comb. Theory, Ser. B , 71(2):130–143, 1997. doi: 10.1006/jctb.1997. 1775

work page doi:10.1006/jctb.1997 1997
[9]

Goedgebeur, J

J. Goedgebeur, J. Renders, and S. Van Overberghe. Cycle Permutation Graphs (Version 2) [Computer software], version 2, 2025. url: https : / / github . com / JarneRenders/Cycle-Permutation-Graphs

work page 2025
[10]

Goedgebeur, J

J. Goedgebeur, J. Renders, G. Wiener, and C. T. Zamfirescu. K2-Hamiltonian Graphs, (Version 1) [Computer software], version 1, 2022. url: https://github. com/JarneRenders/K2-Hamiltonian-Graphs

work page 2022
[11]

Goedgebeur, J

J. Goedgebeur, J. Renders, G. Wiener, and C. T. Zamfirescu. K2-Hamiltonian graphs: II. J. Graph Theory, 105(4):580–611, 2024. doi: 10.1002/jgt.23057

work page doi:10.1002/jgt.23057 2024
[12]

B. Jackson. On circuit covers, circuit decompositions and Euler tours of graphs. In K. Walker, editor, Surveys in Combinatorics, 1993 , London Mathematical Society Lecture Note Series, pages 191–210. Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511662089.008

work page doi:10.1017/cbo9780511662089.008 1993
[13]

V. Klee. Which generalized prisms admit H-circuits? In Y. Alavi, D. R. Lick, and A. T. White, editors, Graph Theory and Applications , volume 303 of Lecture Notes in Mathematics , pages 173–178, Berlin, Germany. Springer, 1972. doi: 10.1007/ BFb0067368

work page 1972
[14]

M´ aˇ cajov´ a and M.ˇSkoviera

E. M´ aˇ cajov´ a and M.ˇSkoviera. Permutation snarks of order 2 (mod 8). Acta Math. Univ. Comen., 88(3):929–934, 2019

work page 2019
[15]

B. D. McKay. Isomorph-free exhaustive generation. J. Algorithms , 26(2):306–324,

work page
[16]

doi: 10.1006/jagm.1997.0898

work page doi:10.1006/jagm.1997.0898 1997
[17]

B. D. McKay and A. Piperno. Practical graph isomorphism, II. J. Symb. Comput. , 60:94–112, 2014. doi: 10.1016/j.jsc.2013.09.003

work page doi:10.1016/j.jsc.2013.09.003 2014
[18]

Pisanski and J

T. Pisanski and J. Shawe-Taylor. Cycle permutation graphs with large girth. Glas. Mat., 17(2):233–236, 1982

work page 1982
[19]

Pisanski and J

T. Pisanski and J. Shawe-Taylor. Search for minimal trivalent cycle permutation graphs with girth nine. Discret. Math., 36(1):113–115, 1981. doi: 10.1016/0012- 365X(81)90179-5

work page doi:10.1016/0012- 1981
[20]

R. C. Read. Every one a winner or how to avoid isomorphism search when catalogu- ing combinatorial configurations. In B. Alspach, P. Hell, and D. J. Miller, editors, Annals of Discrete Mathematics . Volume 2, Algorithmic Aspects of Combinatorics, pages 107–120. Elsevier, 1978. doi: 10.1016/S0167-5060(08)70325-X

work page doi:10.1016/s0167-5060(08)70325-x 1978
[21]

M. Z. Rozenfel’d. The construction and properties of certain classes of strongly regular graphs. Uspehi Mat. Nauk , 28(3):197–198, 1973. In Russian

work page 1973
[22]

P. D. Seymour. Sums of circuits. In J. A. Bondy and U. R. S. Murty, editors, Graph Theory and Related Topics , pages 341–355, New York, NY, USA. Academic Press, 1979

work page 1979
[23]

Szekeres

G. Szekeres. Polyhedral decompositions of cubic graphs. Bull. Aust. Math. Soc. , 8(3):367–387, 1973. doi: 10.1017/S0004972700042660

work page doi:10.1017/s0004972700042660 1973
[24]

W. T. Tutte. A contribution to the theory of chromatic polynomials. Can. J. Math., 6:80–91, 1954. doi: 10.4153/CJM-1954-010-9

work page doi:10.4153/cjm-1954-010-9 1954
[25]

C.-Q. Zhang. Integer flows and cycle covers of graphs , number 205 in Monographs and textbooks in pure and applied mathematics. Marcel Dekker, New York, 1997. 17 A Canonical construction path method In this section we describe how to use the canonical construction path method, which is a generic method for the exhaustive isomorphism-free generation of gra...

work page 1997
[26]

Only accept a graph if it was obtained by canonical expansion

work page
[27]

In our case there is only one expansion operation, that is: adding an edge between pairs of specific vertices of degree 2

For every graph G to which expansions will be applied, only perform one expansion from each equivalence class of expansions. In our case there is only one expansion operation, that is: adding an edge between pairs of specific vertices of degree 2. Let G be a spanning subgraph of a cycle permutation graph which has a consecutive permutation 2-factor F with...

work page
[28]

2”, to “2, 4

of a and of b. • x7 is the number of vertices at distance at most 4 from a or b. 19 We note that while the discriminating power is negligible for the later invariants, they are necessary for the efficiency of variants of this program, such as the generation of non- hamiltonian cycle permutation graphs or cycle permutation graphs with girth restrictions. S...

work page

[1] [1]

Brinkmann and J

G. Brinkmann and J. Goedgebeur. Generation of cubic graphs and snarks with large girth. J. Graph Theory, 86(2):255–272, 2017. doi: 10.1002/jgt.22125

work page doi:10.1002/jgt.22125 2017

[2] [2]

Brinkmann, J

G. Brinkmann, J. Goedgebeur, J. H¨ agglund, and K. Markstr¨ om. Generation and properties of snarks. J. Comb. Theory, Ser. B , 103(4):468–488, 2013. doi: 10.1016/ j.jctb.2013.05.001

work page 2013

[3] [3]

Brinkmann, J

G. Brinkmann, J. Goedgebeur, and B. D. McKay. Generation of cubic graphs. Dis- cret. Math. Theor. Comput. Sci. , 13(2):69–80, 2011. doi: 10.46298/dmtcs.551

work page doi:10.46298/dmtcs.551 2011

[4] [4]

Brinkmann, J

G. Brinkmann, J. Goedgebeur, and B. D. McKay. The minimality of the Georges– Kelmans graph. Math. Comp., 91:1483–1500, 2022. doi: 10.1090/mcom/3701

work page doi:10.1090/mcom/3701 2022

[5] [5]

Chartrand and F

G. Chartrand and F. Harary. Planar permutation graphs. Ann. Inst. H. Poincar´ e, 3(4):433–438, 1967

work page 1967

[6] [6]

Coolsaet, S

K. Coolsaet, S. D’hondt, and J. Goedgebeur. House of Graphs 2.0: a database of interesting graphs and more. Discret. Appl. Math., 325:97–107, 2023. doi: 10.1016/ j.dam.2022.10.013. Available at https://houseofgraphs.org

work page 2023

[7] [7]

I. A. Faradˇ zev. Generation of nonisomorphic graphs with a given degree sequence. In Algorithmic Studies in Combinatorics , pages 11–19. Nauka, Moscow, Russia, 1978. In Russian. 16

work page 1978

[8] [8]

L. A. Goddyn, J. van den Heuvel, and S. McGuinness. Removable circuits in multi- graphs. J. Comb. Theory, Ser. B , 71(2):130–143, 1997. doi: 10.1006/jctb.1997. 1775

work page doi:10.1006/jctb.1997 1997

[9] [9]

Goedgebeur, J

J. Goedgebeur, J. Renders, and S. Van Overberghe. Cycle Permutation Graphs (Version 2) [Computer software], version 2, 2025. url: https : / / github . com / JarneRenders/Cycle-Permutation-Graphs

work page 2025

[10] [10]

Goedgebeur, J

J. Goedgebeur, J. Renders, G. Wiener, and C. T. Zamfirescu. K2-Hamiltonian Graphs, (Version 1) [Computer software], version 1, 2022. url: https://github. com/JarneRenders/K2-Hamiltonian-Graphs

work page 2022

[11] [11]

Goedgebeur, J

J. Goedgebeur, J. Renders, G. Wiener, and C. T. Zamfirescu. K2-Hamiltonian graphs: II. J. Graph Theory, 105(4):580–611, 2024. doi: 10.1002/jgt.23057

work page doi:10.1002/jgt.23057 2024

[12] [12]

B. Jackson. On circuit covers, circuit decompositions and Euler tours of graphs. In K. Walker, editor, Surveys in Combinatorics, 1993 , London Mathematical Society Lecture Note Series, pages 191–210. Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511662089.008

work page doi:10.1017/cbo9780511662089.008 1993

[13] [13]

V. Klee. Which generalized prisms admit H-circuits? In Y. Alavi, D. R. Lick, and A. T. White, editors, Graph Theory and Applications , volume 303 of Lecture Notes in Mathematics , pages 173–178, Berlin, Germany. Springer, 1972. doi: 10.1007/ BFb0067368

work page 1972

[14] [14]

M´ aˇ cajov´ a and M.ˇSkoviera

E. M´ aˇ cajov´ a and M.ˇSkoviera. Permutation snarks of order 2 (mod 8). Acta Math. Univ. Comen., 88(3):929–934, 2019

work page 2019

[15] [15]

B. D. McKay. Isomorph-free exhaustive generation. J. Algorithms , 26(2):306–324,

work page

[16] [16]

doi: 10.1006/jagm.1997.0898

work page doi:10.1006/jagm.1997.0898 1997

[17] [17]

B. D. McKay and A. Piperno. Practical graph isomorphism, II. J. Symb. Comput. , 60:94–112, 2014. doi: 10.1016/j.jsc.2013.09.003

work page doi:10.1016/j.jsc.2013.09.003 2014

[18] [18]

Pisanski and J

T. Pisanski and J. Shawe-Taylor. Cycle permutation graphs with large girth. Glas. Mat., 17(2):233–236, 1982

work page 1982

[19] [19]

Pisanski and J

T. Pisanski and J. Shawe-Taylor. Search for minimal trivalent cycle permutation graphs with girth nine. Discret. Math., 36(1):113–115, 1981. doi: 10.1016/0012- 365X(81)90179-5

work page doi:10.1016/0012- 1981

[20] [20]

R. C. Read. Every one a winner or how to avoid isomorphism search when catalogu- ing combinatorial configurations. In B. Alspach, P. Hell, and D. J. Miller, editors, Annals of Discrete Mathematics . Volume 2, Algorithmic Aspects of Combinatorics, pages 107–120. Elsevier, 1978. doi: 10.1016/S0167-5060(08)70325-X

work page doi:10.1016/s0167-5060(08)70325-x 1978

[21] [21]

M. Z. Rozenfel’d. The construction and properties of certain classes of strongly regular graphs. Uspehi Mat. Nauk , 28(3):197–198, 1973. In Russian

work page 1973

[22] [22]

P. D. Seymour. Sums of circuits. In J. A. Bondy and U. R. S. Murty, editors, Graph Theory and Related Topics , pages 341–355, New York, NY, USA. Academic Press, 1979

work page 1979

[23] [23]

Szekeres

G. Szekeres. Polyhedral decompositions of cubic graphs. Bull. Aust. Math. Soc. , 8(3):367–387, 1973. doi: 10.1017/S0004972700042660

work page doi:10.1017/s0004972700042660 1973

[24] [24]

W. T. Tutte. A contribution to the theory of chromatic polynomials. Can. J. Math., 6:80–91, 1954. doi: 10.4153/CJM-1954-010-9

work page doi:10.4153/cjm-1954-010-9 1954

[25] [25]

C.-Q. Zhang. Integer flows and cycle covers of graphs , number 205 in Monographs and textbooks in pure and applied mathematics. Marcel Dekker, New York, 1997. 17 A Canonical construction path method In this section we describe how to use the canonical construction path method, which is a generic method for the exhaustive isomorphism-free generation of gra...

work page 1997

[26] [26]

Only accept a graph if it was obtained by canonical expansion

work page

[27] [27]

In our case there is only one expansion operation, that is: adding an edge between pairs of specific vertices of degree 2

For every graph G to which expansions will be applied, only perform one expansion from each equivalence class of expansions. In our case there is only one expansion operation, that is: adding an edge between pairs of specific vertices of degree 2. Let G be a spanning subgraph of a cycle permutation graph which has a consecutive permutation 2-factor F with...

work page

[28] [28]

2”, to “2, 4

of a and of b. • x7 is the number of vertices at distance at most 4 from a or b. 19 We note that while the discriminating power is negligible for the later invariants, they are necessary for the efficiency of variants of this program, such as the generation of non- hamiltonian cycle permutation graphs or cycle permutation graphs with girth restrictions. S...

work page