Odd 2–factored snarks
Abstract
A snark is a cubic cyclically –edge connected graph with edge chromatic number four and girth at least five. We say that a graph is odd –factored if for each –factor F of G each cycle of F is odd.
In this paper, we present a method for constructing odd 2–factored snarks. In particular, we construct two new odd –factored snarks that disprove a conjecture by some of the authors. Moreover, we approach the problem of characterizing odd –factored snarks furnishing a partial characterization of cyclically –edge connected odd –factored snarks. Finally, we pose a new conjecture regarding odd –factored snarks.
1 Introduction
All graphs considered are finite and simple (without loops or multiple edges). We shall use the term multigraph when multiple edges are permitted. For definitions and notations not explicitly stated the reader may refer to [10].
A snark (cf. e.g. [24]) is a bridgeless cubic graph with edge chromatic number four (by Vizing’s theorem the edge chromatic number of every cubic graph is either three or four so a snark corresponds to the special case of four). In order to avoid trivial cases, snarks are usually assumed to have girth at least five and not to contain a non–trivial –edge cut (i.e. they are cyclically –edge connected).
Snarks were named after the mysterious and elusive creature in Lewis Caroll’s famous poem The Hunting of The Snark by Martin Gardner in [20], but it was P. G. Tait in 1880 that initiated the study of snarks, when he proved that the four colour theorem is equivalent to the statement that no snark is planar [34]. The Petersen graph is the smallest snark and Tutte conjectured that all snarks have Petersen graph minors. This conjecture was confirmed by Robertson, Seymour and Thomas (cf. [31]). Necessarily, snarks are non–hamiltonian.
The importance of the snarks does not only depend on the four colour theorem. Indeed, there are several important open problems such as the classical cycle double cover conjecture [32, 33], Fulkerson’s conjecture [16] and Tutte’s 5–flow conjecture [35] for which it is sufficient to prove them for snarks. Thus, minimal counterexamples to these and other problems must reside, if they exist at all, among the family of snarks.
Snarks play also an important role in characterizing regular graphs with some conditions imposed on their 2–factors. Recall that a –factor is a –regular spanning subgraph of a graph .
A graph with a –factor is said to be –factor hamiltonian if all its –factors are Hamilton cycles, and, more generally, –factor isomorphic if all its –factors are isomorphic. Examples of such graphs are , , , the Heawood graph (which are all –factor hamiltonian) and the Petersen graph (which is –factor isomorphic). Moreover, a pseudo –factor isomorphic graph is a graphs with the property that the parity of the number of cycles in a –factor is the same for all –factors of . Examples of these graphs are , the Heawood graph and the Pappus graph (cf. [3]). Several papers have addressed the problem of characterizing families of graphs (particularly regular graphs) which have these properties directly [11, 19, 6, 1, 2, 12, 3, 4, 5] or indirectly [17, 27, 28, 18, 29, 7, 15]. In particular, we have recently pointed out in [4] some relations between snarks and some of these families (cf. Section 2).
We say that a graph is odd –factored (cf. [4]) if for each –factor of each cycle of is odd. In [4] we have investigated which snarks are odd –factored and we have conjectured that a snark is odd –factored if and only if is the Petersen graph, Blanuša , or a Flower snark , with and odd (Conjecture 2.5).
At present, there is no uniform theoretical method for studying snarks and their behaviour. In particular, little is known about the structure of –factors in a given snark.
In this paper, we present a new method, called bold–gadget dot product, for constructing odd –factored snarks using the concepts of bold–edges and gadget–pairs over Isaacs’ dot–product [25]. This method allows us to construct two new instances of odd –factored snarks of order and that disprove the above conjecture (cf. Conjecture 2.5). Moreover, we furnish a characterization of bold–edges and gadget–pairs in known odd –factored snarks and we approach the problem of characterizing odd –factored snarks furnishing a partial characterization of cyclically –edge connected odd –factored snarks. Finally, we pose a new conjecture about odd –factored snarks.
2 Preliminaries
Until only five snarks were known, then Isaacs [25] constructed two infinite families of snarks, one of which is the Flower snark [25], for which in [4] we have used the following definition:
Let be an odd integer. The Flower snark (cf. [25]) is defined in much the same way as the graph described in [1].
The graph has vertex set
and edge set
For we call the subgraph of induced by the vertices the interchange of . The vertices and the edges are called respectively the hub and the spokes of . The set of edges linking to are said to be the link of . The edge is called the –channel of the link. The subgraph of induced by the vertices and are respectively cycles of length and and are said to be the base cycles of
The technique used by Isaacs to construct the second infinite family is called a dot product and it is a consequence of the following:
Lemma 2.1 (Parity Lemma)
A dot product (see figure below) of two cubic graphs and , of cyclic–edge–connectivity at least , denoted by is defined as follows [25, 24]:

remove any pair of adjacent vertices and from ;

remove any two independent edges and from ;

join to and to or to and to , where and .
Note that the dot product allows one to construct graphs of cyclic edge–connectivity exactly . Moreover, the dot product of the Petersen graph with itself gives rise to two snarks Blanuša 1 and Blanuša 2.
The Parity Lemma 2.1 allows one to prove the following:
A more general method to construct snarks called superposition has been introduced by M. Kochol [26]. A superposition is performed replacing simultaneously edges and vertices of a snark by suitable cubic graphs with pendant (or half) edges (called superedges and supervertices) yielding a new snark. Superpositions allow one to construct cyclically –edge–connected snarks with arbitrarily large girth, for .
As already mentioned in the Introduction a graph is odd –factored if for each –factor of each cycle of is odd.
By definition, an odd –factored graph is pseudo –factor isomorphic. Note that, odd –factoredness is not the same as the oddness of a (cubic) graph (cf. e.g.[36]).
Lemma 2.3
[4] Let be a cubic –connected odd –factored graph then is a snark.
In [4] some of the authors have posed the following:
Question: Which snarks are odd –factored?
and we have proved:
Proposition 2.4
[4]

Petersen and Blanuša2 are odd 2–factored snarks.

The Flower Snark , for odd , is odd –factored. Moreover, is pseudo –factor isomorphic but not –factor isomorphic.

All other known snarks up to vertices and all other named snarks up to vertices are not odd –factored.
Thus it seemed reasonable to pose the following:
Conjecture 2.5
[4] A snark is odd –factored if and only if is the Petersen graph, Blanuša , or a Flower snark , with and odd.
As mentioned above, the Blanuša graphs arise as the dot product of the Petersen graph with itself, but one is odd –factored (cf. Proposition 2.4(i)) while the other one is not. In the Petersen graph, which is edge transitive, there are exactly two kinds of pairs of independent edges. The Blanuša snarks are the result of these two different choices of the pairs of independent edges in the dot product. We will make use of this property for constructing new odd –factored snarks in Sections 3 and 4.
Proposition 2.6
The dot product preserves snarks, but not odd 2 factored graphs.
3 A construction of odd –factored snarks
We present a general construction of odd –factored snarks performing the dot product on edges with particular properties, called bold–edges and gadget–pairs respectively, of two snarks and .
Construction: Bold–Gadget Dot Product.
We construct (new) odd –factored snarks as follows:

Choose a bold–edge in ;

Choose a gadget–pair , in ;

Perform a dot product using these edges;

Obtain a new odd 2–factored snark (cf. Theorem 3.7).
Note that in what follows the existence of a 2–factor in a snark is guaranteed since they are bridgeless by definition.
Definition 3.1
Let be a snark. A bold–edge is an edge such that the following conditions hold:

All –factors of and of are odd;

all –factors of containing are odd;

all –factors of avoiding are odd.
Note that not all snarks contain bold–edges (cf. Proposition 4.2, Lemma 5.1). Furthermore, conditions and are trivially satisfied if is odd –factored.
Lemma 3.2
The edges of the Petersen graph are all bold–edges.
Proof. Since is hypohamiltonian (i.e. is hamiltonian, for each ) and moreover, for every , all –factors of are hamiltonian , condition holds. The other two conditions are satisfied since is odd–factored.
Definition 3.3
Let be a snark. A pair of independent edges and is called a gadget–pair if the following conditions hold:

There are no –factors of avoiding both ;

all –factors of containing exactly one element of are odd;

all –factors of containing both and are odd. Moreover, and belong to different cycles in each such factor.

all –factors of containing exactly one element of , are such that the cycle containing the new edge is even and all other cycles are odd.
Note that, finding gadget–pairs in a snark is not an easy task and, in general, not all snarks contain gadget–pairs (cf. Lemma 5.2).
Let be the two horizontal edges and the vertical edge respectively (in the pentagon–pentagram representation) of (cf. Figure 1).
It is easy to prove the following properties:
Lemma 3.4
Let be the Petersen graph and be as above.

The graph is bipartite.

The graph has no 2–factors, for any distinct .
Lemma 3.5
Any pair of distinct edges in the set of is a gadget–pair.
Proof. It can be easily checked that the edges of in have the property that any –factor of contains exactly two of them. Moreover, they belong to different cycles of the –factor. Indeed, for any two edges and in their endvertices are all at distance in . Thus the shortest cycle containing both and has length . Since all –factors of contain two –cycles, in any –factor of containing both and , these edges are contained in different cycles. Hence, conditions – follow from the above reasoning and the odd –factoredness of .
Condition can also be easily checked and moreover, any –factor of (or or ), for , containing the new edge is hamiltonian, hence even (and obviously there are no other cycles in these 2–factors).
In the next lemma we recall a well known property of edge–cuts:
Lemma 3.6
Let be a connected graph and let be a set of edges such that is disconnected, but is not disconnected, for any proper subset of . Then, for any cycle of , is even.
Recall that the length of a cycle is denoted by . The following theorem allows us to construct new odd –factored snarks.
Theorem 3.7
Let be a bold–edge in a snark and let be a gadget–pair in a snark . Then is an odd –factored snark.
Proof. Denote , , , , and the 4–edge cut obtained performing the dot product .
Let be a 2–factor of then contains an even number of edges of by Lemma 3.6.
We distinguish three cases according to the number of edges of in :
Case 1. contains no edges of .
In this case it is immediate to check that a subset of the cycles of forms a –factor of , contradicting Definition 3.3(i). Thus there are no 2–factors of avoiding .
Case 2. contains exactly two edges and of .
We want to prove that all cycles of are odd. We distinguish two subcases.
Case 2.1. The endvertices of and in are both endvertices of either or .
W.l.g. we may assume that and . Let and let . Then is a 2–factor of . Analogously, let and let . Then is a 2–factor of containing and avoiding . Let be the cycle of containing . Then is odd by Definition 3.1(i). Similarly, let be the cycle of containing . Then is odd by Definition 3.3(ii). Thus, the cycle of containing and has which is odd. Finally,all other cycles of are odd by Definition 3.1(i) and Definition 3.3(ii).
Case 2.2. The endvertices of and in lie one in and the other in .
W.l.g. we may assume that and . Let and let . Then is a 2–factor of containing . Analogously, let and let . Let be a set of new edges and consider the graph . Then is a 2–factor of containing only of by construction. Let be the cycle of containing . Then is odd by Definition 3.1(ii). Similarly, let be the cycle of containing . Then is even by Definition 3.3(iv). Thus, the cycle of containing and has which is odd. Finally, all other cycles of are odd by Definition 3.1(ii) and Definition 3.3(iv).
Case 3. contains all the four edges of .
Again we want to prove that all cycles of have odd length. Let , , and . Note that is a –factor of avoiding and that is a –factor of containing both and . Let and be the cycles of containing and , respectively. If then we denote such a cycle by . Analogously, let and be the cycles of containing and , respectively. Note that and are always distinct by Definition 3.3(iii).
In order to compute the parity of the length of the cycles of containing , we need to analyze all possible combinations of paths in between the vertices and between the vertices of . It is easy to check that we have five different cases (the others being equivalent to some of these five) but three of them are ruled out by Definition 3.3(iii), since they have (cf. Figure 2).



Case 3.1 
Case 3.2 



These three cases can be ruled out since they give rise to 

in , impossible by Definition 3.3(iii) 
The two remaining subcases are:
Case 3.1. All edges of lie in a cycle of such that and .
Case 3.2. The edges of are contained in two distinct cycles and of such that and .
Thus, the resulting graph is odd –factored, hence a snark by Lemma 2.3.
Construction of
Recall that, the Blanuša2 snark is odd –factored (cf. [4] and Proposition 2.4). We can obtain the same result taking two copies of the Petersen graph , in the first one choosing any edge as a bold–edge (by Lemma 3.2) and in the second a gadget–pair as in Lemma 3.5. The resulting graph, obtained as the dot product , denoted by , is odd –factored by Theorem 3.7 (cf. Figure 4) and it is isomorphic to the Blanuša2 snark.
Let be the two horizontal edges and the vertical edge respectively (in the pentagon–pentagram representation) of the Petersen graph , as in Figure 1. Let and be two copied of . Choose as the bold–edge in and as the gadget–pair in . Moreover, let and be the –poles represented as follows:
Performing the dot–product we obtain the Blanuša2 snark (Figure 4).
Lemma 3.8
Under the above hypothesis, the only bold–edges of are those edges, say and , identified with the edges and of (cf. Figure 4).
Proof. Fix the labelling on as in Figure 4. To find all possible bold–edges in , we only need to verify Definition 3.1, since is odd –factored.
To this purpose, we have implemented a program, with the software package MAGMA [8], and computed that the graph has the dihedral group as automorphism group, its edge–orbits are six and its vertex–orbits are five. For each representative of the vertex–orbits, we have determined all the –factors of (computing the determinant of the variable adjacency matrix of [23]). The only vertex, for which has only odd –factors, is (lying in the vertex–orbit ). Hence, the only bold–edges in are , since there is an edge–orbit of , and its edges correspond to (c.f. Figure 4).
4 Counterexamples to Conjecture 2.5: Constructions
We construct two new examples of odd –factored snarks of order and , denoted respectively as and , and starting from the Petersen and the Blanuša2 snarks applying iteratively the method described in Section 3. These two examples disprove Conjecture 2.5. Moreover, we investigate the structure of the snarks obtained with this method computing their bold–edges and gadget pairs.
Construction of
Proposition 4.1
Let be a copy of and be a copy of . Choose to be one of the two bold–edges in and let be a gadget–pair in . Then the dot product gives rise to a new odd –factored snark . Moreover, the only bold–edge of is , the edge of identified with the edge of (cf. Figure 5).
Proof. Applying the construction given by Theorem 3.7 to the chosen bold–edge (cf. Lemma 3.8) and gadget–pair (cf. Lemma 3.5), we obtain that the graph is an odd 2–factored snark.
Fix the labelling on as in Figure 5. To find all possible bold–edges in , we only need to verify Definition 3.1, since we have just proved that is odd –factored.
To this purpose, as in Proof of Lemma 3.8, we have implemented a program, with the software package MAGMA, and computed that the graph has the dihedral group as automorphism group, its edge–orbits are eight and its vertex–orbits are seven. For each representative of the vertex–orbits, we have determined all the –factors of . The only vertex, for which has only odd –factors, is (lying in the vertex–orbit ). Hence, the only bold–edge in is , since there is an edge–orbit of , and its edge correspond to (c.f. Figure 5).
Construction of
Proposition 4.2
Let be a copy of and be a copy of . Choose to be the only bold–edge in and let be a gadget–pair in . Then the dot product gives rise to a new odd –factored snark . Moreover, has no bold–edges (cf. Figure 6).
Proof. Applying the construction given by Theorem 3.7 to the only bold–edge (cf. Proposition 4.1) and gadget–pair (cf. Lemma 3.5), we obtain that the graph is an odd 2–factored snark.
Fix the labelling on as in Figure 6. To find all possible bold–edges in , again, we only need to verify that Definition 3.1 does not hold, since we have just proved that is odd –factored.
To this purpose, as in Lemma 3.8 and Proposition 4.1, we have implemented a program, with the software package MAGMA, and computed that the graph has the symmetric group as automorphism group, its edge–orbits and its vertex–orbits are both four. For each representative of the vertex–orbits, we have determined all the –factors of . We have obtained that there is always a –factor containing a cycle of even length. Thus, Definition 3.1 does not hold. Hence has no bold–edges.
Remark 4.3
We have learned from J. Hägglund [21] that Brimnkmann, Goedbgebeur, Markstrom and himself had also found in [9], with an exhaustive computer search of all snarks of order , numerical counterexamples to Conjecture 2.5, one of order and one of order , but at the time we have informed him that we had already constructed these counterexamples via the bold–gadget dot product. Indeed, we have checked that the snarks and are isomorphic to their graphs of order and , respectively.
5 A partial characterization of odd –factored snarks
To approach the problem of characterizing all odd –factored snarks, we consider the possibility of constructing further odd –factored snarks with the technique presented in Section 3, which relies in finding other snarks with bold–edges and/or gadget–pairs, Therefore, we study the existence of bold–edges and gadget–pairs in the known odd –factored snarks.
We have already computed all the bold–edges in the Petersen graph , the Blanuša2 snark , and the new snarks and (cf. Lemma 3.2, Lemma 3.8, Proposition 4.1, Proposition 4.2).
Lemma 5.1
Let , for odd , be the Flower Snark. Then has no bold–edges.
Proof. Fix the labelling on the vertices of as defined in Section 2. The flower snark has the dihedral group as automorphism group [13], its edge–orbits are four and its vertex–orbits are three.
To prove that there are no bold–edges, we only need to verify Definition 3.1 does not hold, since we have already proved in [4] that is odd –factored. To this purpose, we have to find a –factor containing an even cycle in , for each representative of the vertex–orbits.
Let and be representatives for the three vertex–orbits of . Then, for each orbit we can construct the following –factor in :
Hence, we have obtained that, for all of these graphs, there is always a –factor containing an even cycle. Thus, Definition 3.1 does not hold. Hence, has no bold–edges.
Regarding gadget–pairs, we have computed so far, only the gadget–pairs in the Petersen graph (cf. Lemma 3.5).
Lemma 5.2
Let the Flower snark , for odd , the Blanuša2 snark , , and be defined as above. Then

, and have no gadget–pairs;

The Flower snark has no gadget–pairs.
Proof. For each of these graphs, we will verify that Definition 3.3 or does not hold.
Fix the labelling on , and as in Figures 4, 5 and 6. For these graphs, we have implemented a program, with the software package MAGMA, in which we compute the edge–orbits under the action of the automorphism group; we consider all independent edges from a chosen representative of each edge–orbit and then we find all –factors of . If any such –factors exist then condition 3.3 does not hold. Otherwise, we choose one of the edges (cf. Definition 3.3), say , then we compute all –factors of and, in each case, we find a –factor for which condition 3.3 does not hold.
In the graphs , and for each representative of one of the edge–orbits, there are several possible independent edges .
For most pairs there exists a –factor of , thus Condition 3.3 does not hold, whereas the pairs for which has no –factors are:
For each of these pairs of edges, admits a –factor in which the cycle has odd length, or has other even cycles besides , contradicting 3.3. Hence, , and have no gadget–pairs, since Definition 3.3 or does not hold.
For the graphs , odd, fix the labelling on the vertices of as defined in Section 2.
Recall that in a cubic graph , a –factor, , determines a corresponding –factor, namely . In studying –factors in it is more convenient to consider the structure of –factors.
If is a –factor of each of the links of contain precisely one edge from This follows from the argument in [1, Lemma 4.7]. Then, a –factor may be completely specified by the ordered –tuple where for each and indicates which edge in belongs to Together these edges leave a unique spoke in each to cover its hub. Note that , (i.e. they lie in different channels, for example if , then ). To read off the corresponding –factor simply start at a vertex in a base cycle at the first interchange. If the corresponding channel to the next interchange is not banned by , proceed along the channel to the next interchange. If the channel is banned, proceed via a spoke to the hub (this spoke cannot be in ) and then along the remaining unbanned spoke and continue along the now unbanned channel ahead. Continue until reaching a vertex already encountered, so completing a cycle At each interchange contains either or vertices. Furthermore as is constructed iteratively, the cycle is only completed when the first interchange is revisited. Since uses either or vertices from it can revisit either once or twice. If revisits twice then is a hamiltonian cycle which is not the case. Hence it follows that consists of two cycles and
Let be independent edges in . Since each of the links of contain precisely one edge from any given –factor of , each –factor of must contain exactly two edges of each link . Therefore, if , for some , then there is no –factor of avoiding both, i.e. Definition 3.3 holds. Hence, to prove statement in this case, we need to verify that Definition 3.3 does not hold. We need first to prove that for all other independent pairs that Definition 3.3 does not hold, namely that contains a –factor. To this purpose we will define a –factor of containing both and , thus giving rise to a –factor of . As noted above, the –factors of can be specified by an ordered –tuple with for . We need to consider the following four cases:
Case 1: belong to different links, i.e. and , with , . Suppose and , for . Choose any –tuple such that and . Define to be the –factor of corresponding to . Note that in the case , , since and are independent.
Case 2: belongs to a link and is a spoke of the same index, i.e. and for some . Suppose and . Choose any –tuple such that and . Define to be the –factor of corresponding to . Since and are independent, .
Case 3: belongs to a link and is a spoke of different index, i.e. and , with , . Suppose and , for . Choose any –tuple such that , , . Define to be the –factor of corresponding to . Moreover, choose and in different channels, which is always possible since there are three channels at each link and only one needs to be avoided.
Case 4: are both spokes in different interchanges, i.e. and , for , . Note that cannot be two spokes in the same interchange since they are independent edges. Suppose and , for . Choose any –tuple such that , , , , which is always possible since there are three channels at each link. Define to be the –factor of corresponding to .
In each case, the –factor corresponding to is well defined and it avoids both and , thus Definition 3.3 does not hold.
This leaves us to prove that in the case for some , in which Definition 3.3 holds, Definition 3.3 does not hold. To this purpose, we choose one of the edges (cf. Definition 3.3), say , and find a –factor for which Definition 3.3 does not hold. Suppose, that and , with , and consider the graph .
Recall that the flower snark has the dihedral group as automorphism group ([13]) with vertex orbits ,