Non-isomorphic Signatures on Some Generalised Petersen Graph

In this paper we find the number of different signatures of $P(3,1), P(5,1)$ and $P(7,1)$ upto switching isomorphism, where $P(n, k)$ denotes the generalised Petersen graph, $2k<n$. We also count the number of non-isomorphic signatures on $P(2n+1,1)$ of size two for all $n \geq 1$, and we conjecture that any signature of $P(2n+1,1)$, upto switching, is of size at most $n+1$.


Introduction
Throughout the paper we consider simple graphs. For all the graph-theoretic terms that have not been defined but are used in the paper, see Bondy [1]. Harary [4] firstly introduced the notion of signed graph and balance. Harary [2] used them to model social stress in small groups of people in social psychology. Subsequently, signed graphs have turned out to be valuable. The fundamental property of signed graphs is balance. A signed graph is balanced if all its cycles have positive sign product. The second basic property of signed graphs is switching equivalence. Switching is a way of turning one signature of a graph into another, without changing cycle signs. Many properties of signed graphs are unaltered by switching, the set of negative cycles is a notable example. In [6], the non-isomorphic signatures on the Heawood graph are studied. The author in [9] determined the non-isomorphic signed Petersen graph, using the fact that the minimum signature on a cubic graph is a matching. Using the same technique, we find the number of non-isomorphic signatures on P (3, 1), P (5, 1) and P (7,1). We also determine the number of non-isomorphic signatures of size two in P (2n + 1, 1) for all n ≥ 1.

Preliminaries
A signified graph is a graph G together with an assignment of + or − signs to its edges. If Σ is the set of negative edges, then we denote the signified graph by (G, Σ). The set Σ is called the signature of (G, Σ). Signature Σ can also be viewed as a function from E(G) into {+1, −1}. A resigning (switching) of a signified graph at a vertex v is to change the sign of each edge incident to v. We say (G, Σ 2 ) is switching equivalent to (G, Σ 1 ) if it is obtained from (G, Σ 1 ) by a sequence of switchings. Equivalently, we say that (G, Σ 2 ) is switching equivalent to (G, Σ 1 ) if there exists a function f : V → {+1, −1} such that Σ 2 (e) = f (u)Σ 1 (e)f (v) for each edge e = uv of G.
Resigning defines an equivalence relation on the set of all signified graphs over G (also on the set of signatures). Each such class is called a signed graph and is denoted by [G, Σ], where (G, Σ) is any member of the class.
We say two signified graphs (G, Σ 1 ) and (H, Σ 2 ) to be isomorphic if there exists a graph isomorphism ψ : V (G) → V (H) which preserve the edge signs. We denote it by Σ 1 ∼ = Σ 2 . They are said to be switching isomorphic if Σ 1 is isomorphic to a switching of Σ 2 . That is, there exists a representation (H, Σ 2 ) which is equivalent to (H, Σ 2 ) such that Σ 1 ∼ = Σ 2 . We denote it by Σ 1 ∼ Σ 2 .
Proposition 2.1. [5] If G has m edges, n vertices and c components, then there are 2 (m−n+c) distinct signed graphs of G.
A cycle in a signified graph (G, Σ) is called positive if the product of its edge signs is positive and negative, otherwise. A signified graph (G, Σ) is called balanced if each cycle in (G, Σ) is positive and unbalanced, otherwise. One of the first theorems in the theory of signed graphs tells that the set of negative cycles uniquely determines the class of signed graphs to which a signified graph belongs. More precisely, we state the following theorem.
Theorem 2.1. [8] Two signatures Σ 1 and Σ 2 of a graph G are equivalent if and only if they have the same set of negative cycles.

Notations
The distance between two vertices x and y in a graph G, denoted by d G (x, y), is the length of a shortest path connecting x and y. The distance between two edges e 1 = u 1 u 2 and e 2 = v 1 v 2 in a graph G, denoted by d G (e 1 , e 2 ), is min{d G (u i , v j ) : i ∈ {1, 2}, j ∈ {1, 2}}. For example, d G (e 1 , e 2 ) = 1 for the edges e 1 = u 0 u 1 and e 2 = u 2 v 2 of the graph P (3, 1) in Figure 1. Throughout this paper, the solid lines and dotted lines in a graph represent positive and negative edges respectively.
In a signed graph [G, Σ], a signature Σ which is equivalent to Σ is said to be a minimum signature if the number of edges in Σ is minimum among all equivalent signatures of Σ. We denote the number of edges in Σ by |Σ |. For example, if [G, Σ] is balanced then Σ = ∅ and thus |Σ | = 0. A signed graph may have more than one minimum signatures. To see this, take a signified complete graph (K 3 , Σ), where Σ = {12, 23, 31}. Switching (K 3 , Σ) by vertex 3 gives an equivalent signature Σ 1 = {12} and by vertex 1 gives an equivalent signature Σ 2 = {23}. Clearly, Σ 1 and Σ 2 are minimum signatures of (K 3 , Σ).
The following theorem tells about the maximum degree of a vertex in a minimum signature, when the signature is considered as a spanning subgraph of a given graph G.
Theorem 3.1. Let [G, Σ] be a signed graph on n vertices and let Σ be an equivalent minimum signature of Σ.
Resign at u to get an equivalent signature Σ 1 . It is clear that |Σ| > |Σ 1 |. We apply the same operation on Σ 1 , if G Σ 1 has a vertex of degree greater than n−1 2 . Repeated application, if needed, of this process will ultimately give us an equivalent signatureΣ of minimum number of edges such that degree of every vertex ofΣ is at most n−1
The following theorem will remain our key result throughout this paper.
Theorem 3.2. [9] Every minimum signature of a cubic graph is a matching.
From now onward, matching of a graph G stands for a minimum signature. With a few exceptions, most of the time switching transforms a matching (when considered as a signature) of P (n, 1) to a new matching. The notation Σ(e 1 , e 2 , . . . , e k ) denotes a signature or a set of edges Σ which contains the edges e 1 , e 2 , . . . , e k of a graph. For example, in the graph P (3, 1) of Figure 1, Σ(u 0 u 1 , v 1 v 2 ) denotes a signature containing the edges u 0 u 1 and v 1 v 2 .
Further, we say that two signatures Σ 1 and Σ 2 of a graph G are automorphic if there exists an automorphism f of G such that uv ∈ Σ 1 if and only if f (u)f (v) ∈ Σ 2 . If two signatures are automorphic then they are said to be automorphic type signatures. If two signatures Σ 1 and Σ 2 of a graph G are not automorphic to each other, then we say that they are distinct automorphic type signatures. For example, in the signed graphs [P (5, 1), {u 1 u 2 }] and [P (5, 1), {u 3 u 4 }], the signatures {u 1 u 2 } and {u 3 u 4 } are automorphic type signatures.

Generalised Petersen Graph
Let n and k be positive integers such that 2 ≤ 2k < n. The generalized Petersen graph, denoted by P (n, k), is defined to have the vertex set {u 0 , u 1 , . . . , u n−1 , v 0 , v 1 , . . . , v n−1 } and edge set where the subscripts are read modulo n. We call the cycle u 0 u 1 . . . u n−1 u 0 as outer cycle and the cycle v 0 v k v 2k . . . v 0 as inner cycle of P (n, k). The edges of the form u i v i are called the spokes of P (n, k). It is clear that P (2n + 1, 1) has 4n + 2 vertices and 6n + 3 edges. Now we discuss certain structural facts of P (2n + 1, 1), where n ≥ 1.
Proof. It is obvious that the cycles given by {u 0 , u 1 , . . . , u 2n } and {v 0 , v 1 , . . . , v 2n } are the only cycles of length 2n + 1. This proves the second part of the theorem.
We prove the first part of the theorem by counting the number of 2l-cycles, where 2 ≤ l ≤ (2n+1). It is important to note that any even cycle in P (2n+1, 1) must contain as many u i 's as v i 's. It is clear that for each i = 0, 1, . . . , 2n, the cycle u i v i v i+1 . . . v i+(l−1) u i+(l−1) u i+(l−2) . . . u i+1 u i is of length 2l, and any even cycle of P (2n + 1, 1) is of this form. Hence there are 2n + 1 cycles of P (2n + 1, 1) of length 2l, where 2 ≤ l ≤ (2n + 1). This proves the theorem.
Theorem 4.2. The distance between any two edges in P (2n + 1, 1) is at most n for all n ≥ 1.
Proof. It is clear that the distance between any two spokes of P (2n+1, 1) is at most n. Further, the distance between any two edges of the outer, as well as of the inner cycle, is at most n. Without loss of generality, if we pick the edge u 0 u 1 from the outer cycle, then the edges v n v n+1 and v n+1 v n+2 are the only edges of the inner cycle which are at maximum distance of n from u 0 u 1 . Similarly, u n+1 v n+1 is the only spoke which is at maximum distance of n from u 0 u 1 . This completes the proof of the theorem.
Accordingly, the automorphisms of P (2n + 1, 1) are some combination of rotations, reflections and interchanges of v i 's with u i 's. Using this fact, if H 1 and H 2 are two given subgraphs of P (2n + 1, 1), it is easier to decide whether there is an automorphism of P (2n + 1, 1) that maps H 1 onto H 2 .
Example 1. The graph P (3, 1) is given in Figure 1. The automorphism ρ 1 rotates the graph P (3, 1) clockwise through the angle 2π 3 , the automorphism δ 1 flips P (3, 1) about the line containing the edge u 1 v 1 as its segment, and γ switches the cycles u 0 u 1 u 2 and v 0 v 1 v 2 to each other.
For more on automorphism group of generalised Petersen graph, see [3]. Yegnanarayanan [7] studied various aspects of the generalised Petersen graph.

Signings on P (3, 1)
From Theorem 3.2, it is easy to see that finding non-isomorphic signatures on P (3, 1) is equivalent to determining the non-isomorphic matchings of P (3, 1) of size up to three. Let M k denotes a matching of size k, where k = 0, 1, 2, 3. We classify all the automorphic type matchings of P (3, 1) of size up to three in the following lemmas. Let a matching of size zero be denoted by Σ 0 .
Lemma 5.1. The number of distinct automorphic type matchings of P (3, 1) of size one is two.
Proof. A matching of size one that does not contain a spoke is Σ 1 (u 0 u 1 ). A matching of size one containing a spoke is Σ 2 (u 0 v 0 ). It is easy to see that any other matching of size one is automorphic to either Σ 1 or Σ 2 , and that Σ 1 is not automorphic to Σ 2 . This proves the lemma.
Lemma 5.2. The number of distinct automorphic type matchings of P (3, 1) of size two is four.
Proof. We classify the matchings of size two by looking at the distance between their edges. Theorem 4.2 gives us that the distance between any two edges of P (3, 1) is at most one.
(i) Let M 2 have no spoke. We may assume that one edge is u 0 u 1 . There are two possibilities for such matchings of size two. One of such matchings is Σ 3 (u 0 u 1 , v 0 v 1 ) and another is Any other matching of P (3, 1) of size two is automorphic to Σ 3 , Σ 4 , Σ 5 or Σ 6 . Further, no two of these matchings are automorphic. This concludes the proof of the lemma. Proof. Any M 3 must contain at least one spoke, as at most one edge can be taken from the inner cycle as well as from the outer cycle. If a matching of size three contains two spokes of P (3, 1), then no other edge can be included in that matching. Thus, following are the possibilities for M 3 .
(i) Let M 3 have one spoke and let it be u 0 v 0 . There is only one possibility for such a matching, and let it be

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Non-isomorphic signatures on some generalised Petersen graph | D. Sehrawat and B. Bhattacharjya (ii) Let M 3 has three spokes and let that Any other matching of size three is automorphic to Σ 7 or Σ 8 , and that Σ 7 is not automorphic to Σ 8 . This completes the proof.
The matchings obtained in the preceding lemmas along with Σ 0 give us nine different automorphic type matchings of P (3, 1) viz., Σ 0 , Σ 1 , . . . , Σ 8 . However, some of these nine matchings may be switching isomorphic to each other. We have the following observations.
So we are left with the matchings Σ 0 , Σ 1 , Σ 2 , Σ 3 , Σ 4 , Σ 5 , and their corresponding signed graphs are depicted in Figure 2, where the label of the vertices correspond to that of Figure 1. In the following theorem we show that these six matchings are not switching isomorphic to each other.
Theorem 5.1. There are exactly six signed P (3, 1) up to switching isomorphisms.
Proof. The number of negative 3-cycles and negative 4-cycles for the signed P (3, 1) shown in Figure 2 are given in Table 1. We see that the set of negative cycles are different for all these six Table 1: Number of negative 3-cycles and negative 4-cycles of some signed P (3, 1).
signatures. So by Theorem 2.1, we conclude that all these six signatures are pairwise not switching isomorphic. This completes the proof. Figure 2: The six signed P (3, 1).
6. Signings on P (5, 1) The graph P (5, 1) is shown in Figure 3. Recall from Theorem 3.2 that finding non-isomorphic signatures of P (5, 1) is equivalent to finding matchings of P (5, 1) of sizes 0, 1, 2, 3, 4 and 5, up to switching isomorphism. We now classify all the automorphic type matchings of P (5, 1) of sizes up to five. We denote a matching of size zero by Σ 0 . We emphasize that at most two edges of a matching may lie on the outer cycle or on the inner cycle. We use this fact to get the possible automorphic type matchings of different sizes. Lemma 6.1. The number of distinct automorphic type matchings of P (5, 1) of size one is two.
Proof. We have only the following two cases.
(i) Let M 1 have no spoke. There is only one automorphic type matching of size one. One of such matchings is There is also only one automorphic type matching of size one. One of such matchings is Any other matching of P (5, 1) of size one is automorphic to Σ 1 or Σ 2 , and that Σ 1 is not automorphic to Σ 2 . This completes the proof.
Lemma 6.2. The number of distinct automorphic type matchings of P (5, 1) of size two is eight.
Proof. We classify the matchings of size two by looking at the distance between the edges of the matching.
(i) Let the edges of the matching be at distance one. There are five different automorphic type matchings of size two and one of each such automorphic type matchings is . Let M 2 be a matching of size two other than Σ 3 , Σ 4 , Σ 5 , Σ 6 and Σ 7 whose edges are at distance one. Note that M 2 must contain either two spokes, or two edges from outer cycle, or two edges from inner cycle, or one edge from outer cycle and one from inner cycle, or one edge from outer/inner cycle and one spoke. In each of these cases, M 2 is automorphic to either Σ 7 , Σ 5 , Σ 3 , Σ 4 or Σ 6 . Thus Σ 3 , Σ 4 , Σ 5 , Σ 6 and Σ 7 are the only automorphic type matchings of size two whose edges are at distance one. It is clear that these matchings are pairwise non-automorphic. (ii) Let edges of M 2 be at distance two. There are three automorphic type matchings of size two whose edges are at distance two. We denote them by In a similar manner (as in case(i)), one can show that any other matching of size two whose edges are at distance two is automorphic to one of Σ 8 , Σ 9 and Σ 10 . The matchings Σ 8 , Σ 9 and Σ 10 are clearly pairwise non-automorphic.
This concludes the proof of the lemma.
Lemma 6.3. The number of distinct automorphic type matchings of P (5, 1) of size three is 11.
Proof. We classify all the automorphic type matchings of size three by looking at the number of spokes contained in these matchings.
. Any other matching of size three containing only one spoke is automorphic to one of Σ 14 , Σ 15 , Σ 16 and Σ 17 . Further, no two of these matchings are automorphic. (iii) Let M 3 have two spokes. If the spokes are consecutive then there is only one possibility, viz., . If the spokes are not consecutive then there is also only one possibility, viz., Σ 19 (u 0 v 0 , v 2 u 2 , u 3 u 4 ). Any other matching of size three containing only two spokes is automorphic to Σ 18 or Σ 19 , and that Σ 18 is not automorphic to Σ 19 . (iv) Let M 3 have three spokes. In this case, there are only two automorphic type matchings of size three and one of each such type of matchings is Σ 20 Any other matching of size three containing only spokes is automorphic Σ 20 or Σ 21 .
This proves the lemma.
Lemma 6.4. The number of distinct automorphic type matchings of P (5, 1) of size four is 10.
Proof. We classify the matchings of size four by considering the number of spokes contained in these matchings.
(i) Let M 4 have no spoke. Note that, at most two edges of M 4 may lie on the outer cycle and at most two edges may lie on the inner cycle. So, without loss of generality, let the edges u 0 u 1 and u 2 u 3 be lie on the outer cycle. Thus following are the only possibilities for the matchings of size four having no spoke: . Any other matching of size four which does not contain a spoke is automorphic to one of Σ 22 , Σ 23 and Σ 24 . Further, these matchings are pairwise non-automorphic. (ii) Let M 4 have one spoke and let it be u 0 v 0 . Out of the three remaining edges, two edges will lie on the outer (inner) cycle and one edge will lie on inner (outer) cycle. The two edges which lie on the outer cycle can be taken to be u 1 u 2 and u 3 u 4 . Thus the possible automorphic type matchings of size four are . Any other matching of size four having only one spoke is automorphic to either Σ 25 or Σ 26 , and that Σ 25 is not automorphic to Σ 26 . (iii) Let M 4 have two spokes. If spokes are at distance one then let they be u 0 v 0 and u 1 v 1 . Further, out of remaining two edges, only one edge may lie on outer cycle and other may lie on inner inner. Let u 2 u 3 lies on outer cycle. Then, the possible automorphic type match- . If the two spokes are at distance two then let they be u 0 v 0 and u 2 v 2 . The only possibility for such matching is . Any other matching of size four with only two spokes is automorphic to one of Σ 27 , Σ 28 and Σ 29 . Also these matchings are pairwise non-automorphic. (iv) Let M 4 have three spokes. If one of the spokes is at distance two from the other two spokes, then no edge from the outer or inner cycle can be contained in M 4 . Therefore, the only possibility is Σ 30 (u 0 v 0 , u 1 v 1 , u 2 v 2 , u 3 u 4 ). (v) Let M 4 have four spokes. One of such matchings is Σ 31 (u 0 v 0 , u 1 v 1 , u 2 v 2 , u 3 v 3 ). Any other matching for this case is automorphic to Σ 31 .
This completes the proof of the lemma. (iii) Let M 5 have five spokes. There is also only one automorphism type of M 5 . We denote it by This completes the proof of lemma.
The matchings obtained in the preceding lemmas along with Σ 0 give us 35 different automorphic type matchings of P (5, 1) viz., Σ 0 , Σ 1 , . . . , Σ 34 . However, some of these 35 matchings may be switching isomorphic to each other. We have the following observations.
edges of P (7, 1). If any one of these seven signatures appears in a matching M l of P (7, 1), where l ≥ 3, then M l is switching equivalent to M l , where l ≤ l − 1.
Proof. Let M j l be a matching of size j which contains the set σ j , where j = 1, . . . , 7 and l ≥ 3. Consider the sets If we resign at the vertices belonging to S j , then we get a signature of size up to l − 1, i.e., M l ∼ M l , where l ≤ l − 1. This proves the lemma. In Lemma 7.1, the inequality l ≤ (l − 1) may be strict. For example, if M 3 = σ 4 , then by resigning at the vertices u 1 , v 1 , u 2 , v 2 , v 3 and u 3 , we get M 3 ∼ M 1 . The matchings σ i , where 1 ≤ i ≤ 7, are said to be forbidden matchings of P (7, 1). Let M 0 denotes the matching Σ 1 = ∅ of size zero. Further, there are only two automorphic type matchings of size one, we denote them by Σ 2 (u 0 u 1 ) and Σ 3 (u 0 v 0 ). Any other matching of P (7, 1) of size one is automorphic to either Σ 2 or Σ 3 , and that Σ 2 is not automorphic to Σ 3 . Lemma 7.2. The number of distinct automorphic type matchings of P (7, 1) of size two is 12.
Proof. We classify these matchings by looking at the distance of their edges. Recall that the distance between any two edges of P (7, 1) is at most three.
(i) Let the edges of M 2 be at distance one. Five such possible matchings of size two are Note that any matching of P (7, 1) of size two contains either two consecutive spokes, or one spoke and one edge from outer (inner) cycle or two edges from the outer (inner) cycle, or one edge from the inner cycle and one from the outer cycle. Each such possible M 2 , whose edges are at distance one, is automorphic to one of Σ 8 , Σ 6 , Σ 7 , Σ 4 and Σ 5 . These five matchings are also pairwise non-automorphic. (ii) Let the edges of M 2 be at distance two. There are only four automorphic type matchings of size two having edges at distance two. We denote them by . It is easy to see that any other matching of size two whose edges are at distance two is automorphic to one of Σ 9 , Σ 10 , Σ 11 and Σ 12 . Further, these matchings are pairwise non-automorphic. (iii) Let the edges of M 2 be at distance three. There are only three automorphic type matchings of size two whose edges are at distance three. We denote them by Σ 13 . Any other M 2 whose edges are at distance three is automorphic to one of Σ 13 , Σ 14 and Σ 15 . Further, no two of these matchings are automorphic to each other.
This completes the proof.
Lemma 7.3. The number of distinct automorphic type matchings of P (7, 1) of size three is 23.
Proof. We classify matchings of size three on the basis of the number of spokes contained in it. Since each forbidden matching is a matching of size three and that they are switching equivalent to a matching of size at most two, we consider matchings other than the forbidden matchings.
(i) Let M 3 have no spoke. The possible distinct automorphic type matchings of size three without spokes are denoted by Σ 16 (u 0 u 1 , u 2 u 3 , u 4 u 5 ), Σ 17 (u 0 u 1 , u 2 u 3 , v 4 v 5 ) and Σ 18 (u 0 u 1 , u 4 u 3 , v 5 v 6 ). Any other matching of size three with no spokes is either a forbidden matching or automorphic to one of Σ 16 , Σ 17 and Σ 18 . Further, it is easy to see that they are pairwise non-automorphic. (ii) Let M 3 have only one spoke, say u 0 v 0 . Possible automorphic type matchings are Any other matching of size three containing only one spoke is automorphic to one of these twelve matchings. Further, any two of these matchings are pairwise non-automorphic. (iii) Let M 3 have only two spokes. If the spokes are consecutive then let they be v 0 u 0 and v 1 u 1 . Thus the only possible automorphic type matching is Σ 31 (v 0 u 0 , v 1 u 1 , u 3 u 4 ). If spokes are at distance two then let the spokes be v 0 u 0 and v 2 u 2 . The possible automorphic type matchings are Σ 32 (v 0 u 0 , v 2 u 2 , u 3 u 4 ) and Σ 33 (v 0 u 0 , v 2 u 2 , u 4 u 5 ). If the spokes are at distance three then let they be v 0 u 0 and v 3 u 3 . The possible automorphic type matchings are Σ 34 (v 0 u 0 , v 3 u 3 , u 1 u 2 ) and Σ 35 (v 0 u 0 , v 3 u 3 , u 4 u 5 ). Because of the forbidden matchings and the automorphism group of P (7, 1), it is easy to see that any other matching of size three containing only two spokes is automorphic to one of Σ 31 , Σ 32 , Σ 33 , Σ 34 and Σ 35 . Further, these matchings are pairwise non-automorphic. (iv) Let M 3 have three spokes. The possible automorphic type matchings of size three having three spokes are Any matching of size three with three spokes is automorphic to Σ 36 , Σ 37 or Σ 38 . Also, these matchings are pairwise non-automorphic.
This completes the proof of the lemma. Further, these three matchings are pairwise non-automorphic. (iii) Let M 4 have two spokes. If the spokes are consecutive then by resigning at the end vertices of the spokes lying on the outer cycle, we find that the matching is equivalent to a matching (signature) of size four. Note that this resultant signature is either equivalent to a matching of size four of case (i) or it is not a matching. Therefore in both cases, it is switching equivalent to a matching M l , where l ≤ 3. If the spokes are at distance two or three, then the possible automorphic type matchings of size four are Σ 42 . Any other matching of size four containing only two spokes is automorphic to one of these six matchings. Further, any two of these matchings are pairwise non-automorphic. (iv) Let M 4 have three spokes. It is clear that a matching of size four cannot contain two or three consecutive spokes up to switchings. Thus the only possible automorphic type matching is Σ 48 (v 0 u 0 , v 2 u 2 , v 4 u 4 , u 5 u 6 ).
(v) Let M 4 have four spokes. By resigning at the vertices from the set {u 0 , u 1 , u 2 , u 3 , u 4 , u 4 , u 6 }, we see that such a matching is equivalent to a matching of size three containing three spokes.
This proves the lemma.
Theorem 7.1. Every matching of P (7, 1) of size five is switching equivalent to a matching M l , where l ≤ 4.
Proof. We prove the theorem by classifying the matchings of size five on the basis of number of spokes contained in it.
(i) Let M 5 have no spoke. Note that three edges of M 5 may lie on the outer (inner) cycle and remaining two edges lie on the inner (outer) cycle. It is easy to see that each such combination of five edges of P (7, 1) must contain either σ 2 or σ 4 . Therefore by Lemma 7.1, each such M 5 is switching equivalent to a matching M l , where l ≤ 4. Note that in (a), the three edges lying on the outer cycle must be u 1 u 2 , u 3 u 4 and u 5 u 6 . Therefore for the fifth edge of M 5 , whichever edge we choose from the inner cycle, M 5 will contain either σ 2 or σ 4 . Hence by Lemma 7.1, each such M 5 is switching equivalent to a matching M l , where l ≤ 4. In (b), let two edges of M 5 lying on the outer cycle be u 1 u 2 and u 3 u 4 . The possibilities for the remaining two edges from the inner cycle are In all of these possible cases, it is easy to see that M 5 contains a forbidden matching. Hence by Lemma 7.1, each such M 5 is switching equivalent to some matching M l , where l ≤ 4. In the similar way, it can be proved that any other matching of size five, containing one spoke and two edges from both outer as well as inner cycles, is switching equivalent to a matching M l , where l ≤ 4. (iii) Let M 5 have two spokes. If M 5 has two consecutive spokes, i.e., spokes at distance two, then by switching at their end vertices lying on outer cycle, we find that M 5 is switching equivalent to either a signature of P (7, 1) of size five with no spokes or a matching of size five with no spoke. In both cases, M 5 is switching equivalent to a matching M l , where l ≤ 4. If two spokes of M 5 are at distance two then let they be u 0 v 0 and u 2 v 2 . The remaining three edges have to be chosen from {u 3 u 4 , u 4 u 5 , u 5 u 6 , v 3 v 4 , v 4 v 5 , v 5 v 6 }. We can take two edges from outer cycle and one edge from inner cycle or vice-versa. If the two edges from the outer cycle are u 3 u 4 and u 5 u 6 , then for all possible choices of the edge from the inner cycle, M 5 must contain either σ 1 or σ 2 . Similarly, if two edges are taken from the inner cycle, M 5 will contain some forbidden matching. Hence each such M 5 is switching equivalent to a matching M l , where l ≤ 4. Similarly, it can be shown that if the two spokes of M 5 are at distance three, then also M 5 is switching equivalent to a matching M l , where l ≤ 4.  Figure 6: Twenty seven signed P (7, 1).
Further, the number of negative 4-cycles in M 11 2 , M 12 2 , M 13 2 and M 14 2 are 0, 3, 2 and 2, respectively. The number of negative 2n + 1-cycles in M 11 2 , M 12 2 , M 13 2 and M 14 2 are 2, 1, 2 and 0, respectively. These numbers of negative cycles show that M 11 2 , M 12 2 , M 13 2 and M 14 2 are pairwise non-isomorphic. Hence up to switching isomorphism, M 11 2 , M 12 2 , M 13 2 and M 14 2 are the only matchings of size two whose edges are at distance one. Similarly, For each i, one can show up to switching isomorphism that there are exactly four matchings of size two whose edges are at distance i, where 2 ≤ i < n.
Three matchings of size two whose edges are at distance n are M n1 It is clear that any other matching of size two whose edges are at distance n is automorphic to one of M n1 2 , M n2 2 and M n3 2 . Further, the number of negative 4-cycles in M n1 2 , M n2 2 and M n3 2 are 3, 2 and 4, respectively. Thus these three matchings are pairwise non-isomorphic.
It is easy to see that any two matchings of size two whose edges are at different distances are different up to switching isomorphisms. This concludes the proof of theorem.
Having proved this theorem, we propose the following problem.
In Section 5, we noticed that P (3, 1) has no matching (or minimum signature) of size 3, up to switching isomorphism. In Section 6, we noticed that P (5, 1) has no matching of sizes 4 and 5, up to switching isomorphism. In Section 7, we noticed that P (7, 1) has no matching of sizes 5, 6 and 7, up to switching isomorphism. On the basis of these observations, we propose the following conjecture.
Conjecture 1. Any matching in P (2n + 1, 1) can be of size at most n + 1, where n ≥ 4. Now Problem 2 can be reformulated as follows.