On dispersability of some circulant graphs

The matching book thickness of a graph is the least number of pages in a book embedding such that each page is a matching. A graph is dispersable if its matching book thickness equals its maximum degree. Minimum page matching book embeddings are given for bipartite and for most non-bipartite circulants contained in the (Harary) cube of a cycle and for various higher-powers.


Introduction
Dispersable graphs were introduced in [4], where it was conjectured that all regular bipartite graphs are dispersable.This was disproved by Alam et al. [1] who showed that the Gray and Folkman graphs, though regular bipartite, are not dispersable.These counterexamples are edge-transitive but not vertex-transitive.In [2], Alam et al. gave an infinite family of counterexamples to the claim and conjectured that bipartite vertex-transitive graphs are dispersable.
Matching book embeddings of bipartite circulants C are given where the page number is equal to the vertex degree ∆(C), supporting the conjecture [2].Regular dispersable graphs are bipartite [21].A nonbipartite circulant is nearly dispersable if it needs one extra page [21].So far, all nonbipartite circulants have been nearly dispersable and we conjecture that nonbipartite, vertextransitive graphs are nearly dispersable.
Previous results support both conjectures.For the complete bipartite graph and the hypercube, see [4]; for complete graphs and other bipartite graphs, see [21].Cartesian products of even cycles are dispersable; even times odd cycles are nearly dispersable [17]; and short odd (length at most 5) and arbitrary odd cycles have nearly dispersable product [15].Other classes of vertex-transitive graph that are known to be nearly dispersable include the product of two arbitrary cycles and of cycles with complete graphs, see [23,25], and some products of bipartite and nonbipartite graphs [22].See also [27] and §8.Some graphs which are not vertex transitive also are dispersable such as trees [21], Halin trees [24], and cubic planar bipartite graphs [2,19].
To define good matching book embeddings for an infinite family of graphs, one needs to give both layout and coloring schemes: algorithms which produce the needed vertex order and edgeto-page assignment from the various integers that identify each graph in the family.Most of our families consist of circulants C(n, S) with a fixed jump-length set S and with the number n of vertices reduced modulo 2 or 4. The coloring algorithms can either be static (as in tables based on modularity) or dynamic (as in prescriptions for Hamiltonian cycles or paths).See proofs of Theorems 1 and 3, resp.
It turns out, however, that for nonbipartite circulants, perfectly regular patterns almost never succeed and irregularity is forced.Irregular features appear in two different ways: local and global.
The local type of exception is involved in the "twist" (see §3) while the global type manifests as "sparseness" in many examples, where one nearly reaches the lower bound except for a "sparse" page with a small and structurally defined set of exceptional edges.Computer search [18, p 7] gives random vertex-order, while our vertex-orders and edge-to-page functions are quite regular.Nevertheless, the edges of the sparse page are irregularly distributed in a characteristic pattern for all parameter values with the same modularity.
Any strategy to achieve the minimum number of pages in a matching book embedding of a graph family, such as C(n, S), based on regular layout and algorithms is a kind of "polymerization process" since almost all edges are placed in a repeated pattern.A polymer is a molecule composed of a sequence of many parts such as proteins composed of amino acids or RNA/DNA as a sequence of nucleotides.The sequence of parts may form a path or a cycle.
Here are four examples of generalized polymerization, a phenomenon that we believe deserves more thorough investigation.In each of these examples, a finite set of adjustments permits regularity for the arbitrarily large remainder.
(2) Layouts and page-partition of the Cartesian product of C 3 or C 5 with another cycle [15] use a "seed" that is an exceptional copy of a repeated motif.
(4) Most of the matching book embeddings in this paper have a sparse page.In contrast, strict polymerization is defined below to be an algorithmic procedure which puts together certain modular units with no adjustments.

Definitions
Undefined terms are as in [11].
The circulant graph C(n, S) of order n with jump set S = {i 1 , . . ., i k } is the graph on [n] := {1, . . ., n}, where j ∈ [n] is adjacent to j + i r (addition mod n), r = 1, . . ., k and 1 A graph is vertex-transitive if for any two vertices, there is an isomorphism carrying one to the other.The k-th Harary power C k n of an order-n cycle C n [11, p 14] is the circulant C(n, [k]), and any circulant of order n with maximum jump k is a vertex-transitive subgraph of C k n .The cube of a cycle is the 3rd power.A drawing of a graph is outerplane (or convex or circular) if its vertices are placed along a circle (or the boundary of any convex region) and the edges are straight lines.
Two edges in an outerplane drawing cross if they intersect at a non-endpoint.Let (G, ω) denote the outerplane drawing of a graph G with cyclic order ω on V (G).
A book embedding [4] of a graph G is an outerplane drawing and an edge-partition such that edges in the same part do not cross.The parts of the partition are the pages of the book embedding.The book thickness bt(G) of G is the least number of pages in any book embedding while bt(G, ω) is the least number of pages for the outerplane drawing (G, ω).
A proper edge-coloring c of a graph G is a function c : E(G) → {1, . . ., r} (the set of colors) such that adjacent edges get different colors.Let χ ′ (G) be the least number of colors in a proper edge-coloring.The remarkable theorem of Vizing [11, p 133] A matching book embedding is a book embedding where the pages are matchings (no two edges are adjacent).The matching book thickness of a graph is the least number of pages in any matching book embedding; we write mbt(G) or mbt(G, ω) as for book thickness.If c is the edge-coloring determined by the pages, then the matching book embedding is the triple (G, ω, c).
Clearly, for every graph G, we have ∆(G) ≤ χ ′ (G) ≤ mbt(G).A matching book embedding (G, ω, c) is dispersable if the number |c| of colors equals ∆(G) and is nearly dispersable [15] if |c| = 1 + ∆(G).A graph is dispersable if it has a dispersable embedding and is nearly dispersable if it is not dispersable and has a nearly dispersable embedding.If G is regular and dispersable, then it is bipartite [21].The sparseness s(G, ω, c) of a nearly dispersable book embedding is the least number of edges on any page.The sparseness s(G) of a nearly dispersable graph G is the minimum sparseness over all minimum-page matching book embeddings.
Lemma 1 Let G be a regular nearly dispersable graph of order n.Then the sparseness of G is at least 1 if n is even and at least ∆/2 if n is odd.
Proof: For n even, this is in Overbay [21], while for n odd, each page has at least one uncovered vertex, so for any set of ∆ pages, there is a set of ≥ ∆ distinct points which need to be covered by edges from the remaining page. □ For each n, the page with s edges is the sparse or "exceptional" page [18]; cf.[8].
It remains for us to define the m-fold polymerization of a matching book embedding of a circulant to form a circulant with the same set of jump-lengths but m-fold more vertices with no increase in the number of pages.
For n ≥ 7 and nonempty S ⊆ {1, 2, . . ., ⌈ n−1 2 ⌉}, let (C(n, S), ν n , c), ν n := (1, . . ., n), be a matching book embedding.We call an edge e = a i a k ∈ where d C (u, w) denotes the C n -distance between two vertices u and w, where C n is the graph induced by the cyclic vertex order.The sets E Λ and E Σ of long and short edges form a nontrivial partition of E. If 1 ∈ S, then the edges a i a i+1 are short for i = 1, . . ., n − 1 but the edge a 1 a n is long.If n = 8 and 3 ∈ S, then a i a i+3 ∈ E s for 1 ≤ i ≤ 5, while a 6 a 1 , a 7 a 2 , and a 8 a 3 are long.
Proof: Place m copies of the vertex set of C(n, S) from left to right, where, for the j-th copy, the vertices a j 1 , . . ., a j n are placed from left to right.Thus, is a list of the nm vertices in the m copies of C(n, S).Put in all the short edges for all the copies.For j = 1, . . ., m − 1, each long edge a j i a j k with k > i is replaced by a j k a j+1 i and each long edge One obtains C(nm, S) with vertex order ν nm .Use the same coloring c for the edges in C(nm, S) as in the matching book embedding for C(n, S).Long and short edges cross in the j-th copy if and only if their images under the edge-rearrangement cross correspondingly.Hence, c is a page assignment.□ This process defines the m-fold strict polymerization of the circulant and of its matching book embedding.Note that equality can fail to hold in (1) -e.g., for an even polymerization of an odd cycle.If (C(n, S), ν n , c) is dispersable, then so is C(nm, S), ν nm , c), while if (C(n, S), ν n , c) is nearly dispersable, then C(nm, S), ν nm , c) is either dispersable or nearly dispersable.If both are nearly dispersable, then s(C(nm, S), ν nm , c) ≤ m • s(C(n, S), ν n , c).
By definition, the edges on each of these pages are pairwise-disjoint, while pages are crossingfree since the edges in a color class can only be (i) non-crossing edges of the common twist, (ii) isolated edges on the outer cycle of the form 4−5 or (n−1)−n, (iii) nested edges of the form {a−(a+3), (a+1)−(a+2)}.
Assign each of the four colors red, purple, green, and blue to 2k edges and the fifth color, black, to the two remaining edges as follows: Black: 1−2, 3−n (sparseness is 2; as ∆ = 4, this is the minimum).
Since the edge pairs of the form {a−(a+3), (a+1)−(a+2)} are nested for each value of a, which increases in increments of four for a fixed color, it is clear that no two edges of the same color cross.As a takes on all values from 1 to 4k, these pairs of nested edges cover 8k distinct edges of the graph.In the case where a = 4k + 1 = n, the edge pair {a−(a+3), (a+1)−(a+2)} is {n−3, 1−2}, reducing mod n; these two edges fit on the fifth page without crossing.
This process, illustrated with k = 3 for the graph C(13, {1, 3}) in Fig. 4, may be viewed as taking the red coloring and rotating it three more times, switching colors for each rotation.Now add a fifth color for the last two edges.
Proof: We color this circulant in three cases using paths and cycles.
For n ≥ 5 odd, draw the circulant using the odd-up, even-down cyclic order  As above, color the edges of the zig-zag path 1, 2, 3, . . ., n using the colors red, blue alternatingly.If n ≡ 0 (mod 4), color the two disjoint non-self-crossing cycles n−1, n−3, . . ., 1, n−1 and 2, 4, . . ., n, 2 (both of length n/2) alternating purple and green.This accounts for 2n − 1 edges and the remaining edge 1−n is colored black.The sparseness is 1 which is the minimum possible for an even order, nearly dispersable graph.See Fig. 5, center.
The orange and aqua pages each consist of a total of r non-crossing parallel edges.The red and blue pages also each contain r non-crossing edges with two parallel edges on each page through the center and the remaining r−2 edges on the outer boundary.The remaining two black edges, which is the minimum possible number, clearly do not intersect on the sparse page since the r−(r+2) edge lies on the outer cycle and does not share an endpoint with 2−(n−1).Hence, all 4r + 2 = 2n edges are accounted for.□ Theorem 6 For n ≥ 7 odd, C(n, {1, 2, 3}) is nearly dispersable w.r.t.ω n .

Proof:
Let n = 2r+1 ≥ 7. Use the identical layout and coloring scheme as in Theorem 5.This will cover all distance 2 and distance 3 edges.Now observe that a purple-green non-crossing Hamiltonian path 1, 2, 3, . . ., n can be added to cover all of the distance-1 edges, with the exception of edge 1−n.This last edge can be placed on the black (sparse) page and does not intersect either r−(r+2) or 2−(n−1).We note that this new cycle contributes r purple edges, r green edges, and 1 black edge as shown in Fig. 7 left.Combining this with C(n, {2, 3}), we have accounted for all 6r + 3 = 3n edges of C(n, {1, 2, 3}) and have achieved an optimal sparseness of 3 = ∆/2.See Fig. 7 right for the combined nearly-dispersable coloring of C(n, {1, 2, 3}) for n odd.□ It is natural to ask if the above scheme for adding distance-1 edges allows even values of n.This almost works, with the exception of the black edge 1−n, which intersects edges on the sparse page in the above layout for even values of n, so for even n, mbt(C(n, {1, 2, 3}) − e) = 7, where e = 1−n.

Larger degree and jump-lengths
We now give minimum layouts of C(n, {1, 2, 3}) for some even values of n and for a variety of circulants with ∆ > 3, using polymerization and periodicity.Proof: Use Lemma 2 on the embedding (K r , ν r , c) in [21]; see Fig 8 and 9.
□  Corresponding to the n = 2k case, for k ≥ 3, the cocktail party graph O k := K 2k − kK 2 is the complement of a 1-factor.It is also the 1-skeleton of the n-dimensional octahedron and is regular with ∆O k = 2k − 2. As it contains triangles, the octahedron is at best nearly dispersable.
We show that it does have a nearly dispersable embedding, with 2k − 1 pages, but the embedding does not use the standard vertex ordering and so we don't have a direct way to polymerize it.However, the natural vertex order gives a matching book embedding with one additional page and this can be polymerized.For the n = 2k + 1 case, the same things can be done with C n , the complement of C n .
Proof: For n ≥ 6, even or odd, the complete graph K n is nearly dispersable using the natural vertex order with pages being the 1-factors produced by maximal families of parallel edges given in [21, p 87]; see Fig. 10 and Fig. 12.By our remark above about how the edges can correspond to various jump-lengths in the isomorphic circulant graph, we note that for n = 2k and the folded order ϕ 2k , the set of length-k edges, E k , are a parallel matching, one of the 2k pages in the nearly dispersable matching book embedding of K 2k , and the removal of E k leaves a 2k − 1-page layout of O k .See right side of Fig. 11.
For n = 2k +1, the length-k edges in the circulant constitute a Hamiltonian cycle Z.The folded order takes this 2k + 1-cycle into 2 of the 2k + 1 pages of the standard matching book embedding of K 2k+1 given in [21], with one extra edge.Deleting Z gives a 2k − 1-page layout of C 2k+1 (Fig.
The next theorem allows multiples of 12.
Proof: An explicit coloring of the edges with respect to natural order ν n using 7 = 1 + ∆ colors is given as follows: Periodically, 4-color the edges of length 1, and for every edge of length 3, use the same color as the unique edge of length 1 with which it is nested, thus 4-coloring all edges of length 1 or 3. Three new colors (periodically) suffice for the remaining edges.See Figure 11: A perfect matching in K 8 .

Discussion
Recently, Yu, Shao and Li [27] have shown dispersability (or near dispersability) for circulants of degree 3 and degree 4 with all jump-lengths according to whether or not they are bipartite.This extends our results for these degrees.Our methods supply different solutions to the problem of finding optimal page-number matching book embeddings for such circulants.
We have also considered higher degree circulants and have analyzed some of the structural features of matching book embeddings of regular graphs.
For C(n, {1, 2, 3}), we show near dispersability when n is odd and when n is even and divisible by 7 or 12.We also show near dispersability for the circulants resulting by deleting a maximum matching from an even-order K n and a spanning cycle from an odd-order K n .The analogous results holds in the bipartite case.
Results are constructive, not just existential, and so will remain useful even if the full conjecture on vertex-transitive graphs is proved (or disproved).Some condition on the graph is needed as a regular graph can have an arbitrarily large value  for the ratio mbt/∆ according to Alam et al. [2], which uses a counting argument of McKay [20] to prove that, for any fixed ∆ ≥ 3, there exist ∆-regular bipartite graphs G n with mbt(G n ) → ∞ as n → ∞.But vertex-transitive graphs are rather special.Du, Kutnar & Marušič [7] showed that the Lovasz conjecture (Every vertex transitive graph contains a Hamiltonian cycle, with five exceptional cases) is correct when the order is a product of two primes and the graph satisfies additional conditions involving the action of a group.Also, Diestel [6, p 52; Ex 12] notes that every connected, even-order, vertex-transitive graph has a 1-factor.
In many cases, our proof of near dispersability for a family of matching book embeddings uses a sparse page with the minimum number of edges.Indeed, the lower bound of 2 is achieved in the proof of Theorem 2 for C(n, {1, 3}) when n ≡ 1 (mod 4) while we get sparseness s ≤ 3 for n ≡ 3 (mod 4).The proof of Theorem 3 shows C(n, {1, 2}) = 2 for n odd, while for n ≡ 0 (mod 4), sparseness = 1 is achieved but for n ≡ 2 (mod 4), we only get sparseness ≤ 2. The proof of Theorem 4 shows s(C(n, {2, 3})) ≤ 4 if n is even.Sparseness has minimum value (2 and 3, resp.)for Theorems 5 and 6 when n odd and for degree 6.Our matching book embeddings for Theorems 7, 8 and 9, in contrast, are quite symmetric and so the opposite of sparse embeddings.
For a nearly dispersable embedding, sparseness allows the deletion of a small number of edges to eliminate an entire page, while symmetry might be preferred for an "online" problem where the graph being embedded is evolving.
Matching book thickness for regular graphs has a clear lower bound, so a coloring which achieves the minimum is detectable.We think that finding the matching book thickness of various graphs could be a good target for genetic algorithms, neural networks, or artificial intelligence.See, for example, [3].Application of machine learning techniques to matching book thickness might improve computational theory and practice; see, e.g., [13].Indeed, one has an endless supply of vertex-transitive graphs on which to test procedures.

Figure 1 :Figure 2 :
Figure 1: Common four-coloring c of the twist