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Electronic Journal of Combinatorics 17(1) (2010), R32. \href{https://doi.org/10.37236/304}{\nolinkurl{doi:10.37236/304}}.} \expandafter\def\csname paper@GZ18\endcsname{% Jan Goedgebeur and Carol T. Zamfirescu. \emph{On hypohamiltonian snarks and a theorem of Fiorini}. Ars Mathematica Contemporanea 14(2) (2018), 227--249. \href{https://doi.org/10.26493/1855-3974.1176.9b7}{\nolinkurl{doi:10.26493/1855-3974.1176.9b7}}.} \expandafter\def\csname paper@MS19\endcsname{% Edita M\'a\v{c}ajov\'a and Martin \v{S}koviera. \emph{Permutation snarks of order $2\pmod 8$}. Acta Mathematica Universitatis Comenianae 88(3) (2019), 929--934. \href{https://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1313}{journal version}.} \expandafter\def\csname paper@MS25\endcsname{% Edita M\'a\v{c}ajov\'a and Martin \v{S}koviera. \emph{Are there any permutation snarks on $6\pmod 8$ vertices?} Procedia Computer Science 273 (2025), 163--168. \href{https://doi.org/10.1016/j.procs.2025.10.294}{\nolinkurl{doi:10.1016/j.procs.2025.10.294}}.} \expandafter\def\csname paper@BGHM13\endcsname{% Gunnar Brinkmann, Jan Goedgebeur, Jonas H\"agglund and Klas Markstr\"om. \emph{Generation and properties of snarks}. Journal of Combinatorial Theory, Series B 103(4) (2013), 468--488. \href{https://doi.org/10.1016/j.jctb.2013.05.001}{\nolinkurl{doi:10.1016/j.jctb.2013.05.001}}.} \expandafter\def\csname paper@NS95\endcsname{% Roman Nedela and Martin \v{S}koviera. \emph{Atoms of cyclic connectivity in cubic graphs}. Mathematica Slovaca 45(5) (1995), 481--499. \href{https://dml.cz/dmlcz/132921}{DML-CZ:132921}.} \newcommand{\paperreference}[1]{\csname paper@#1\endcsname} \makeatother \title[Graph Puzzles IV.1-preview: Factorisations of snarks]{Graph Puzzles IV.1-preview: Factorisations of hypohamiltonian and permutation snarks} \author{Nikolay Ulyanov + AI-Slop} \date{6 October 2026} \hypersetup{ pdftitle={Graph Puzzles IV.1-preview: Factorisations of hypohamiltonian and permutation snarks}, pdfauthor={Nikolay Ulyanov + AI-Slop}, colorlinks=true, linkcolor=blue!50!black, citecolor=blue!50!black, urlcolor=blue!50!black } \begin{document} \selectlanguage{english} \maketitle Nikolay Ulyanov\textsuperscript{*} + AI-Slop% \marginnote{\textsuperscript{*}{\scriptsize\href{mailto:ulyanick@gmail.com}{ulyanick@gmail.com}}}\par \begin{abstract} \noindent\newthought{Abstract: } Every hypohamiltonian snark and every permutation snark has a unique multiset of cyclically $5$-edge-connected factors of the same class under successive decompositions along cycle-separating $4$-edge-cuts. We prove that hypohamiltonicity passes from a dot product to both of its snark factors. Together with the factorisation theorem for bicritical snarks, this answers a question of Goedgebeur and Zamfirescu. For permutation snarks, we extend the atom argument of Chladn\'y and \v{S}koviera using colourable cut multipoles and closure under their canonical completions. The two conclusions also hold within the class of hypohamiltonian permutation snarks. \end{abstract} \noindent\rule{\linewidth}{0.4pt} \Needspace{12\baselineskip} \section{Introduction} The dot product combines two snarks across a $4$-edge-cut. Reversing this operation repeatedly expresses a snark in terms of factors with higher cyclic connectivity. Chladn\'y and \v{S}koviera~\bothcite{CS10} proved that a bicritical snark has a unique multiset of cyclically $5$-edge-connected bicritical factors. Their theorem identifies the isomorphism types of the factors and their multiplicities, independently of the cuts chosen during the decomposition. A graph is \emph{hypohamiltonian} if it is non-Hamiltonian and every single-vertex deletion is Hamiltonian. Hypohamiltonian snarks are bicritical. Goedgebeur and Zamfirescu~\bothcite{GZ18} asked whether every hypohamiltonian snark of cyclic connectivity four is a dot product of two hypohamiltonian snarks \citep[Problem~4.3]{GZ18}. We prove that both snark factors of any hypohamiltonian dot product are hypohamiltonian. The unique factorisation theorem for bicritical snarks then gives the following result. \begin{restatable}{theorem}{hypofactorisation}\label{thm:hypo} Every hypohamiltonian snark has a unique multiset, up to isomorphism and with multiplicities, of cyclically $5$-edge-connected hypohamiltonian snark factors. Every intermediate factor in a decomposition along cycle-separating $4$-edge-cuts is hypohamiltonian. \end{restatable} A \emph{permutation $2$-factor} consists of two induced cycles. A snark admitting such a $2$-factor is a \emph{permutation snark}. Its two cycles have equal odd length, and their complement is a perfect matching between them. M\'a\v{c}ajov\'a and \v{S}koviera~\bothcite{MS19} proved that every cycle-separating $4$-edge-cut of a permutation snark determines two smaller permutation snarks \citep[Section~2 and Theorem~2.1]{MS19}. Permutation snarks are conjectured to be bicritical; see M\'a\v{c}ajov\'a and \v{S}koviera~\bothcite{MS25}, Section~1. Together with the factorisation theorem for bicritical snarks, this conjecture makes it natural to ask whether permutation snarks also have unique factorisations. We use the closure property above to prove the following result. \begin{restatable}{theorem}{permfactorisation}\label{thm:perm} Every permutation snark has a unique multiset, up to isomorphism and with multiplicities, of cyclically $5$-edge-connected permutation snark factors. Every intermediate factor in a decomposition along cycle-separating $4$-edge-cuts is a permutation snark. \end{restatable} Permutation snarks include non-hypohamiltonian graphs. There is a $26$-vertex example obtained as a successive dot product of three Petersen graphs, and there is also a $50$-vertex example with cyclic connectivity five. Brinkmann, Goedgebeur, H\"agglund and Markstr\"om~\bothcite{BGHM13} enumerated $64$ permutation snarks on $26$ vertices \citep[Table~3]{BGHM13}. Thus the hypohamiltonian permutation snarks form a proper subclass of the permutation snarks, also within the cyclically $5$-edge-connected class. The two factorisation theorems apply to the same completed factors at each cut, giving the following result for their intersection. \begin{restatable}{corollary}{intersectionfactorisation}\label{cor:intersection} Every hypohamiltonian permutation snark has a unique multiset, up to isomorphism and with multiplicities, of cyclically $5$-edge-connected hypohamiltonian permutation snark factors. Every intermediate factor has both properties. \end{restatable} \Needspace{12\baselineskip} \section{Dot products and cut multipoles}\label{sec:preliminaries} All graphs are finite. We use \emph{snark} for a simple cubic graph of girth at least five, cyclically $4$-edge-connected, and with chromatic index four. A \emph{cycle} is a connected $2$-regular subgraph. For $X\subseteq V(G)$, write $\partial_G X$ for the edges with exactly one end in $X$. The cut is \emph{cycle-separating} when both $G[X]$ and $G[V(G)\setminus X]$ contain a cycle. We write $\lambda_c(G)$ for cyclic edge-connectivity, $g(G)$ for girth, and $|G|$ for $|V(G)|$. \subsection{The dot product} Let $A$ and $B$ be disjoint snarks. Choose independent edges $ab,cd$ of $A$ and an edge $xy$ of $B$, with \[ N_B(x)=\{a',b',y\},\qquad N_B(y)=\{c',d',x\}. \] The four vertices $a',b',c',d'$ are distinct. Put \[ L=A-\{ab,cd\},\qquad R=B-\{x,y\}, \] where edges are deleted from $A$ and vertices from $B$. The dot product with this attachment is \begin{equation}\label{eq:dot-product} G=A\mathbin{\cdot}B =L\cup R+\{aa',bb',cc',dd'\}. \end{equation} The four joining edges form the \emph{bond} $F$. Its two couples are $\{aa',bb'\}$ and $\{cc',dd'\}$. This is the convention of \citep[Section~2]{CS10} and \citep[Section~2]{GZ18}. The order satisfies \begin{equation}\label{eq:order} |G|-2=(|A|-2)+(|B|-2). \end{equation} A cycle-separating $4$-edge-cut in a snark is a matching. Indeed, if a vertex is incident with at least two cut edges, moving it to the other shore reduces the size of the cut and preserves a cycle in each shore. The resulting cut has size at most three, contrary to cyclic $4$-edge-connectivity. Thus each shore has four distinct boundary vertices. \subsection{Canonical completions} Disconnecting the four edges of a cut leaves a \emph{$4$-pole} on each shore: a cubic multipole with four semiedges. A proper $3$-edge-colouring of a multipole assigns colours to its ordinary edges and semiedges, with three distinct colours incident at each vertex. We use the multipole terminology of \citep[Section~3]{CS10}. Up to a permutation of colours, the Parity Lemma permits four types of boundary colouring: \[ 1111,\qquad1122,\qquad1212,\qquad1221. \] Every colourable $4$-pole admits at least two of these types. If two colourable $4$-poles join to a snark, their sets of types are disjoint, so each admits exactly two. Such a pole is \emph{colour-open}. Its four semiedges have a unique partition into two couples. In the \emph{isochromatic} case, the semiedges of each couple receive the same colour in every colouring. In the \emph{heterochromatic} case, the semiedges of each couple receive distinct colours in every colouring. The two poles of the cut have opposite types, and their couples agree across the cut \citep[Propositions~3.1--3.5]{CS10}. The \emph{canonical snark completion} of a heterochromatic pole joins the semiedges within each couple. For an isochromatic pole, it attaches each couple to a new vertex and joins the two new vertices. These completions are unique up to isomorphism. They recover the two factors in \eqref{eq:dot-product}, with the heterochromatic completion playing the role of $A$ and the isochromatic completion the role of $B$. A \emph{composition chain} of $G$ is a multiset obtained from $\{G\}$ by successively replacing a member with its two canonical factors at a cycle-separating $4$-edge-cut. A chain is \emph{terminal} when every member is cyclically $5$-edge-connected. Two chains are \emph{equivalent} if their members can be matched, including multiplicities, by graph isomorphisms. We use the terminology of \citep[Section~7]{CS10}. \Needspace{12\baselineskip} \section{Hypohamiltonian factors}\label{sec:hypohamiltonian} We first record the boundary restrictions imposed by the non-Hamiltonicity of the two factors. \Needspace{9\baselineskip} \begin{lemma}\label{lem:boundary} In \eqref{eq:dot-product}, let $A$ and $B$ be non-Hamiltonian graphs. A spanning path in $L$ with terminal ends joins different pairs among $\{a,b\}$ and $\{c,d\}$. A spanning path in $R$ with terminal ends joins $a'$ to $b'$ or $c'$ to $d'$. Two disjoint paths spanning $L$, with ends $a,b,c,d$, pair $a$ with $b$ and $c$ with $d$. Two disjoint paths spanning $R$, with ends $a',b',c',d'$, pair $a'$ with $b'$ and $c'$ with $d'$. \end{lemma} \begin{proof} A spanning $a$--$b$ path in $L$ closes to a Hamiltonian cycle of $A$ when $ab$ is restored; the same holds for $c,d$. A spanning path in $R$ whose ends belong to different pairs closes through $x,y$ to a Hamiltonian cycle of $B$. For a spanning pair of paths in $L$, either cross pairing of the four ends becomes one Hamiltonian cycle after restoring $ab$ and $cd$. For a spanning pair in $R$, either cross pairing becomes one Hamiltonian cycle after adjoining $a'xb'$ and $c'yd'$. The remaining pairing is the one stated in each case. \end{proof} \Needspace{9\baselineskip} \begin{lemma}\label{lem:descent} If a dot product $G=A\mathbin{\cdot}B$ of snarks is hypohamiltonian, then both $A$ and $B$ are hypohamiltonian. \end{lemma} \begin{proof} Use the notation of \eqref{eq:dot-product}. A Hamiltonian cubic graph is $3$-edge-colourable, so both factors are non-Hamiltonian. A Hamiltonian cycle in any vertex-deleted subgraph of $G$ meets $F$ in two or four edges. With two crossings, its restriction to the undeleted shore is a spanning path; with four, that restriction consists of two disjoint spanning paths. Fix $v\in V(A)$ and take a Hamiltonian cycle $C$ in $G-v$. If $C$ uses two bond edges, \cref{lem:boundary} shows that its spanning path in $R$ has ends $a',b'$ or $c',d'$. Replace the whole segment through $R$, including its bond edges, with $ab$ or $cd$, respectively. If $C$ uses all four bond edges, its two spanning paths in $R$ pair $a'$ with $b'$ and $c'$ with $d'$. Replace these two segments with $ab$ and $cd$. Each replacement preserves the cyclic order of the remaining vertices of $C$. The resulting cycle spans $A-v$. Thus $A-v$ is Hamiltonian for every $v\in V(A)$. Now fix $w\in V(B)\setminus\{x,y\}$ and take a Hamiltonian cycle in $G-w$. With two bond edges, the spanning path in $L$ joins different terminal pairs. For example, a path from $a$ to $c$ is replaced, together with its bond edges, by $a'xyc'$. The other choices give the corresponding path through $x,y$. With four bond edges, the two spanning paths in $L$ pair $a$ with $b$ and $c$ with $d$. Replace them, together with their bond edges, by $a'xb'$ and $c'yd'$. The resulting cycle spans $B-w$ in either case. It remains to recover Hamiltonian cycles after deleting $x$ or $y$. Take a Hamiltonian cycle of $G-a$. The missing bond edge $aa'$ forces exactly two crossings. Its spanning path in $R$ has ends $c',d'$ by \cref{lem:boundary}. Adjoining $c'yd'$ gives a Hamiltonian cycle of $B-x$. A Hamiltonian cycle of $G-c$ similarly has a spanning $a'$--$b'$ path in $R$; adjoining $a'xb'$ gives a Hamiltonian cycle of $B-y$. Hence both factors are hypohamiltonian. \end{proof} A snark is \emph{bicritical} if deleting any two distinct vertices yields a $3$-edge-colourable graph. Hypohamiltonian snarks have this property \citep[Section~1]{GZ18}. For distinct $u,v$, deleting $v$ from a Hamiltonian cycle of $G-u$ leaves a spanning path in $G-\{u,v\}$. Colour the path alternately with colours $1,2$. The remaining edges form a matching and receive colour $3$. \Needspace{11\baselineskip} \hypofactorisation* \begin{proof} Let $G$ be hypohamiltonian and let $S$ be any cycle-separating $4$-edge-cut. By bicriticality and \citep[Theorem~7.1]{CS10}, $S$ determines a dot product $G=A\mathbin{\cdot}B$ of smaller bicritical snarks. By \cref{lem:descent}, both factors are hypohamiltonian. Repeating the argument keeps every factor hypohamiltonian. Each decomposition replaces a graph with two smaller graphs, so every chain has a terminal refinement. Its members are cyclically $5$-edge-connected hypohamiltonian snarks. The bicritical unique-factorisation theorem \citep[Theorem~C, or Theorem~10.3]{CS10} identifies any two such terminal chains up to isomorphism and order, including multiplicities. \end{proof} In particular, every hypohamiltonian snark of cyclic connectivity four is a dot product of two hypohamiltonian snarks. This gives an affirmative answer to \citep[Problem~4.3]{GZ18}. \Needspace{12\baselineskip} \section{A factorisation criterion}\label{sec:criterion} The permutation result uses the atom argument for bicritical snarks \citep[Sections~8--10]{CS10}. We formulate the argument with the closure hypotheses that enter its proof. Let $\mathcal C$ be a class of snarks closed under isomorphism. Assume that every cycle-separating $4$-edge-cut in a member of $\mathcal C$ has two colourable shore poles, and that both canonical completions belong to $\mathcal C$. For a cyclic fragment $X$, write $\widetilde X$ for the completion of its shore pole and $G/\widetilde X$ for the completion of the complementary pole. Suppose $\lambda_c(G)=4$. A \emph{cyclic fragment} is an induced shore of a cycle-separating $4$-edge-cut, and an \emph{atom} is a cyclic fragment minimal under inclusion. Since $4