Matching book thickness of generalized Petersen graphs

The matching book embedding of a graph G is to place its vertices on the spine, and arrange its edges on the pages so that the edges in the same page do not intersect each other and the edges induced subgraphs of each page are 1-regular. The matching book thickness of G is the minimum number of pages required for any matching book embedding of G , denoted by mbt ( G ) . In this paper, the matching book thickness of generalized Petersen graphs is determined.


Introduction
The book embedding of a graph was first introduced by Bernhart and Kainen [1]. The book embedding problem of a graph consists of arranging the vertices on the spine (a line) in order and placing the edges in the pages (the half planes with the spine as the common boundary), so that the edges in the same page do not intersect each other. The book thickness of a graph G is the least number of pages that G can be embedded under all vertex orders, denoted by bt(G). On the basis of book embedding, an additional condition is added that the edge induced subgraphs on each page are 1-regular, which is called matching book embedding. The matching book thickness of G is the minimum number of pages that G can be matching embedded, denoted by mbt(G). The graph G is said to be dispersable if it has a matching book embedding in ∆(G) pages, i.e. mbt(G) = ∆(G), where ∆(G) is the maximum degree of G.
The book embedding of graphs have numerous applications(see [2][3]). Reference [4][5][6][7] and its references study the book embedding of some graphs, and obtain the book thickness of graphs. But there are few conclusions about the matching book thickness of graphs, especially the exact value. Bernhart and Kainen [1] proposed that complete bipartite graphs K n,n (n ≥ 1), even cycles C 2n (n ≥ 2), binary n-cube Q(n) (n ≥ 0) and trees are dispersable and Overbay [8] gave detailed proof. For cartesian product graphs, Kainen [9] and Shao, Liu and Li [10] obtained that the matching book thickness of C m × C n (m is even) and K m × C n (m, n ≥ 3), respectively. In 2021, Kainen and Overbay showed that cubic planar bipartite graphs are dispersable (see [11]). For a Halin graph H, Shao, Geng, and Li [12] proved that mbt(H) = 4, if ∆(H) = 3 and The generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. These graphs were first defined by Watkins [13]. The generalized Petersen graph P (n, k), n ≥ 3 and 1 ≤ k ≤ n − 1, has vertex-set {i, i | i = 1, 2, . . . , n} and edge-set The generalized Petersen graph P (n, k) is always a 3-regular graph with 2n vertices and 3n edges, and P (5, 2) is the well known Petersen graph, it can be matching embedded in a 4-page book (see Figure 1). From the definition of P (n, k), P (n, k) ∼ = P (n, n − k) for k < n 2 , thus we can assume 1 ≤ k < n 2 . In this paper, we obtain that the matching book thickness of a generalized Petersen graph P (n, k) as follows.

The Proof of Main Theorem
The proof will be completed by a sequence of lemmas.  Suppose X = {x 1 , . . . , x i } is an ordered vertex set with i vertices in P (n, k), we define X := {x 1 , . . . , x i } and X −1 := {x i , . . . , x 1 }, and call them paired ordering of X and reverse ordering of X, respectively. Lemma 2.3. If P (n, k) is a generalized Petersen graph, n is even and 1 ≤ k < n 2 , then mbt(P (n, k)) = 3, k is odd, 4, k is even.
Proof. According to the parity of k, we need to consider two cases to prove the matching book thickness of P (n, k).
Case 2. k is even. In this case, P (n, k) is 3-regular but not bipartite because it contains at least one odd cycle. According to Lemma 2.2, P (n, k) is not dispersable, mbt(P (n, k)) ≥ ∆(P (n, k)) + 1 = 4. Let d = gcd(n, k) be the greatest common denominator of n and k (here n and k are even, so d is even), σ i is an ordered vertex set and We put all vertices on the spine in the ordering , (σ 1 ) and assign the edges of P (n, k) to 4 pages as follows. Page Therefore we get a matching book embedding of P (n, k) in a 4-page book, mbt(P (n, k)) ≤ 4. (See Figure 2(b) for the case k = 4 and n = 10).
Proof. Since n is odd, P (n, k) contains an odd cycle C n according to its definition, it is not bipartite, and then P (n, k) is 3-regular, so it is not dispersible by Lemma 2.2, mbt(P (n, k)) ≥ ∆(P (n, k)) + 1 = 4.
Let n = ak +r (a ≥ 2, 0 ≤ r < k), According to the parity of a, we need to consider two cases to prove that mbt(P (ak + r, k)) ≤ 4 as follows.
Case 1. a is even. It is easy to know that r is odd because n is odd and a is even. Assume . . , B a−1 , B −1 a ; ϕ 1 = {1, 2, . . . , k − r, α 1 , α −1 2 }. Now we can construct a matching book embedding of P (ak + r, k) in a 4-page book as follows. Put all 2n vertices on the spine in the ordering ϕ 1 (ϕ −1 1 ) and assign all edges to 4 pages.
Case 2. a is odd.
In this case, we prove mbt(P (ak + r, k)) ≤ 4 from the following two situations.
(1) k is even and r is odd.
All 2n vertices of P (ak+r, k) are assigned on the spine by the ordering of ϕ 2 (ϕ −1 2 ) and all edges of P (ak + r, k) can be matching embedded in a 4-page book as follows.
(2) k is odd and r is even.
We can construct a matching book embedding of P (ak + r, k) in a 4-page book as follows. All the vertices are placed on the spine in the order ϕ 3 (ϕ −1 3 ) , and all the edges can be placed in 4 pages so that the edges of the same page do not intersect each other and the edges induced subgraphs of each page are 1-regular.
So P (ak + r, k) can be matching embedded into 4 pages.
In summary, we can get a matching book embedding of P (ak + r, k) in in a 4-page book. Hence mbt(P (ak + r, k)) ≤ 4. (See Figure.3(d) for the case a = 3, k = 5 and r = 2 ) The result is obtained.

Conclusions
In this paper, we study the matching book embedding of the generalized Petersen graph P (n, k), and give different matching book embedding manners for the different parity of n and k. Finally, we determine the matching book thickness of P (n, k), that is, mbt(P (n, k)) = 3 when n is even and k is odd, and mbt(P (n, k)) = 4 in other cases.

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Matching book thickness of generalized Petersen graphs | Zeling Shao et al.