Tight Factorizations of Girth-$ G $-Regular Graphs

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2 Egc girth-3-regular graphs

3 Theorem 1. [22] There is only one (e1)(e2)(e3)-graph Γ with (e1)(e2)(e3)=222=2 . More- 2 over, this Γ is egc. All other proper (e1)(e2)(e3)-graphs are 1 0-graphs, but not necessarily egc.

Proof. This is as the proof of item (1) of Theorem 5.1 in [22]. In order to determine which 120-graphs are egc, let Γ′ =(V ′, E′,φ′) be a finite undirected cubic multigraph. Let e ∈ E′ with φ′(e) = {u, v} and u, v ∈ V ′. Then, e determines two arcs (that is, ordered pairs of end-vertices of e) denoted (e; u, v) and (e; v,u). (If the girth g(Γ′) of Γ′ is larger than 2, then Γ′ is a simple graph, a particular case of multigraph). The following definition is an adaptation of a case of the definition of generalized truncation in [14]. Let A′ denote the set of arcs of Γ′. A vertex-neighborhood labeling of Γ′ is a function ρ : A′ →{1, 2, 3} such that for each u ∈ V ′ the restriction of ρ to the set A′(u)= {(e; u, v) ∈ A′ : e ∈ E′; φ′(e) = {u, v}; v ∈ V ′} of arcs leaving u is a bijection. For our purposes, we require ρ(e; u, v)= ρ(e; v,u), ∀e ∈ E′ with φ′(e)= {u, v}, so that each e ∈ E′ is assigned a well-defined color from the color set {1, 2, 3}. This yields a 1-factorization of Γ′ with three ′ ′ ′ ′ 1-factors that we can call E1, E2, E3 for respective color 1, 2, 3, with E being the disjoint ′ ′ ′ union E1 ∪ E2 ∪ E3. For the sake of examples in Fig. 1, to be presented below, let colors 1,2 and 3 be taken as red, blue and green, respectively. Let K3 be the triangle graph with vertex set {v1, v2, v3}. The triangle-replaced graph ′ ′ ′ ∇(Γ ) of Γ with respect to ρ has vertex set {(ei; u, vi): u ∈ V ;1 ≤ i ≤ 3} and edge set

′ {(ei; u, vi)(ej; u, vj)|vivj ∈ E(K3)}∪{u, vρ(e;u,w))(w, vρ(e;w,u))|e ∈ E ; φ(e)= {u,w}}.

′ 2 Note that ∇(Γ ) is a 1 0-graph. We will refer to the edges of the form (ei; u, vi)(ej; u, vj) as ′ ′ ∇-edges, and to the edges (ei; u, vi)(ej; w, vj), u =6 w, as Γ -edges. Observe that a Γ -edge is incident only to ∇-edges and that each vertex of ∇(Γ′) is incident to precisely one Γ′-edge. This yields the following observation.

Lemma 2. [14] Let Γ′ be a ﬁnite unidirected cubic multigraph of girth g. Then, for any vertex-neighborhood labeling of Γ′, the shortest cycle in the triangle-replaced graph ∇(Γ′) containing a Γ′-edge is of length at least 2g.

2 We say that Γ′ is a generalized snark if its chromatic index χ′(Γ′) is larger than 3. Two examples of generalized snark are: (i) the Petersen graph and (ii) the multigraph obtained by joining two (2k + 1)-cycles (k ≥ 1) via an extra-edge (a bridge between the two (2k +1)- cycles) and adding k parallel edges to each of the two (2k + 1)-cycles so that the resulting ′ ′ multigraph is cubic, see Fig. 1(r) for k = 5. The triangle-replaced graph Γ1 = ∇(Γ ) of a generalized snark Γ′ will also be said to be a generalized snark, as well as the triangle- ′ ′ ′ ′ replaced graph Γi+1 = ∇(Γi) of Γi, for i =1, 2,..., etc. We will say that Γ is snarkless if it is not a generalized snark.

Figure 1: Producing (e1)(e2)(e3)-graphs that are egc

Vertex-neighborhood labelings of the examples of Γ′ presented below and represented in Fig. 1 are indicated with the elements 1, 2 and 3 of ρ(A′) interpreted respectively as edge colors red, blue and green. The smallest snarkless Γ′’s are: (a) the cubic multigraph ′ ′ Γa of two vertices and three edges as in Fig. 1(a), with ∇(Γa) as the triangular prism P rism(K3) = K2K3, as in Fig. 1(e), where V (K2) = {0, 1} and stands for the graph ′ cartesian product [19]; (b) the cubic multigraph Γb of four vertices resulting as the edge-

3 ′ disjoint union of a 4-cycle and a 2-factor 2K2 as in Fig. 1(b), with ∇(Γb) as in Fig. 1(j). Given a snarkless Γ′, a new snarkless multigraph Γ′′ is obtained from Γ′ by replacing any edge e with end-vertices say u, v, by the submultigraph resulting as the union of a path ′ ′ ′ ′ ′ P4 =(u,u , v , v) and an extra edge with end-vertices u , v . For example, Γ as in Fig. 1(a) has Γ′′ as in Fig. 1(s) and ∇(Γ′′) as in Fig. 1(t). Using this replacement of an edge e by the ′ ′′ said submultigraph, one can transform Γb with the enclosed blue edge e into a Γb as in Fig. ′′ ′′ 1(c), with ∇(Γb ) as in Fig. 1(k); or with the two enclosed red edges into a Γb as in Fig. ′′ 1(d), with ∇(Γb ) as in Fig. 1(l). The replacing submultigraphs are also shown enclosed in Fig. 1(k) and Fig. 1(l). ′ ′ The triangle-replaced graph ∇(Γ ) of any snarkless (e1)(e2)(e3)-graph Γ , either proper 3 2 3 2 or improper, with (e1)(e2)(e3) ∈{2 , 1 0, 0 }, yields an egc 1 0-graph, as illustrated via the ′ ′ four graphs in Fig. 1(e-h), namely Γ = ∇(Γa),K4,K3,3 and the 3-cube graph Q3, onto the four 120-graphs Γ in Fig. 1(m-p). This also raises the observation that non-equivalent ′ ′ ′ 1-factorizations F, F of an (e1)(e2)(e3)-graph Γ , like in Fig. 1(h-i) for Γ = Q3, result in non-equivalent 1-factorizations ∇(F ), ∇(F ′)of Γ = ∇(Γ′), represented in this case on ′ Γ= ∇(Γ )= ∇(Q3) in Fig. 1(p-q). This leads to the ﬁnal assertion in Theorem 3. Fig. 1(u) is the egc 120-graph given by ∇(Γ′) for the dodecahedral graph Γ′, in which the union of any two edge-disjoint 1-factors of a 1-factorization of Γ′ yields a Hamilton cycle. In contrast, the Coxeter graph Γ′ in Fig. 1(v) is non-hamiltonian, but the union of any two of its (edge-disjoint) 1-factors is the disjoint union of two 14-cycles, whose apparent interiors are shaded yellow and light gray in the ﬁgure. Thus, Γ = ∇(Γ′) is an egc 120-graph.

Theorem 3. A 120-graph Γ is egc if and only if Γ is the triangle-replaced graph of a snarkless Γ′. Moreover, non-equivalent 1-factorizations of such Γ′ result in corresponding non- equivalent 1-factorizations of Γ.

Proof. There are two types of edges in a 120-graph Γ, namely the triangle edges (those belonging to some triangle of Γ) and the remaining non-triangle edges. Each vertex v of Γ is incident to a unique non-triangle edge ev and is nonadjacent to a unique edgee ¯v (opposite to v) in the sole triangle Tv of Γ to which v belongs. In any 1-factorization F =(F1,...,Fg) of Γ, both ev ande ¯v belong to the same factor Fi (i =1,...,g). Moreover, each edge e = {u, v} of Γ (where e = eu = ev) belongs solely to corresponding triangles Tu and Tv with opposite edgese ¯u ande ¯v. Clearly, {e = eu = ev, e¯u, e¯v} ⊆ Fi with equality given precisely when Γ is the triangular prism as in Fig. 1(e). We will deﬁne the inverse operator ∇−1 of ∇ that applies to each egc 120-graph Γ. Given one such Γ, contracting simultaneously all the triangles T of Γ consists in removing the edges T T T of those T and then identifying the vertices v1 , v2 , v3 of each T into a corresponding single T vertex vT , where vi , for i ∈{1, 2, 3}, has its unique incident non-triangle edge of Γ possessing ′ T color i. This is done so that whenever two triangles T and T have respective vertices vi T ′ T T ′ and vj adjacent in Γ (i, j ∈ {1, 2, 3}), then i = j and the edge vi vi of Γ is removed and replaced by a new edge vT vT ′ . The result of these simultaneous triangle contractions is a ′ ′ ′ ′ ′ ′ multigraph Γ = (V , E ,φ ) with each vT ∈ V incident to three edges of E , one per each color in {1, 2, 3}. The ensuing edge coloring in Γ′ corresponds to a vertex-neighborhood labeling ρ : A′ →{1, 2, 3} of Γ′, from which it follows that Γ is the triangle-replaced graph of

4 Γ′ with respect to ρ, that is: ∇−1(Γ) = Γ′. This establishes an identiﬁcation of Γ and ∇(Γ′) so that the triangle-edges of Γ are the ∇-edges of ∇(Γ′), and the non-triangle edges Γ are the Γ′-edges of ∇(Γ′). This implies the main assertion of the statement of the theorem.

3 Egc girth-4-regular graphs

In this section and in Section 4, we consider (e1)(e2)(e3)(e4)-graphs with (e1)(e2)(e3)(e4) =6 1111 (before considering 1111-graphs in Section 5).

Figure 2: Examples of (e1)(e2)(e3)(e4)-graphs

Many such graphs are toroidal and obtained from the square tessellation denoted by its Schl¨aﬂi symbol {4, 4}. As in [28] Section 6, “Let T be the group of translations of the plane

5 that preserve {4, 4}. Then T is isomorphic to Z × Z and acts transitively on the vertices of {4, 4}. If U is a subgroup of finite index in T , then M = {4, 4}/U is a finite map of type {4, 4} on the torus, and every such map arises this way”. A symmetry α of {4, 4} acts as a symmetry of M if and only if α normalizes U. Every such M has symmetry group Aut(M) transitive on vertices, horizontal edges and vertical edges. Moreover, for each edge e of M, there is a symmetry that reverses e. This yields a fundamental region (of the tessellation or lattice, see [7, 8]) that we call a cutout, denote by Φ and define as a rectangle r squares wide, t squares high, with the left and right edges identified by parallel translation, and the bottom edges identified with the top edges after a shift of s squares to the right, as in Fig. 6 of [28]. A toroidal graph with such a cutout s will be denoted {4, 4}r,t. While the aim of [23, 24, 28] is the study of edge-transitive graphs, s we find 1-factorizations of g-tight graphs in graphs {4, 4}r,t of a more ample nature. Our notation for the vertices of those cutouts will be (i, j), or ij if no confusion arises, where 0 ≤ i 6) with generating sets n {±1, ±i} (1 4), i.e. the lexicographic products of an n-cycle and the complement K2 of K2; and (iii) the bipartite complement of the Heawood graph, treated in Subection 3.3; (iv) for a g-regular graph Γ, the subdivided double DΓ[23, 28] of Γ is the bipartite graph with vertex set (V (Γ) × Z2) ∪ E(Γ) and an edge between vertices (v, i) ∈ V (Γ) × Z2 and e ∈ E(Γ) whenever v is incident to e in Γ; Lemma 4.2 [23] asserts that if Γ is 4-regular and arc-transitive, then DΓ is 4-regular and semisymmetric; for example, the Folkman graph (Fig. 2(a)) is the subdivided double DK5 of the complete graph K5. Theorem 4. The following 34-, 3222- and 24-graphs exist, and are egc or not as indicated: 1. 34-graphs comprehending the: (a) bipartite complement of the Heawood graph, which is not egc, (Subsection 3.3); 0 (b) 4-cube graph Q4 = {4, 4}4,4, which is egc in two different, orthogonally related ways, (Subsection 3.1; an initial example is in Fig. 2(d);)

(c) 4-regular subdivided doubles DΓ, circulant graphs Cn(i, j) and wreath graphs W (n, 2), all with egc-obstructions formed by three edge-disjoint paths of length

6 2 between two nonadjacent vertices; (e.g. D(K5), C10(1, 3) and W (6, 2) in Fig. 2(a-c), with egc-obstructions formed by four green edges and two red edges;)

2. toroidal 2232-graphs (assuming 0 < t ≤ r and 0 ≤ s

0 Z 0 (a) {4, 4}2ℓ,4: egc ⇔ ℓ ∈ (3, ∞) ∩ 2 ; (concatenating copies of {4, 4}4,4 in Fig. 2(d);) s Z Z s 5 7 (b) {4, 4}4s,1: egc ⇔ s ∈ (4, ∞) ∩ \ 2 ; (Fig. 2(h-i) for rt = 201, 281;) 3. toroidal 24-graphs (assuming 0 < t ≤ r and 0 ≤ s

s Z Z Z Z s 7 (a) {4, 4}r,1: egc ⇔ (r, s) ∈ (2 \ 4 ) × ( \ (2 ∪ 1)); (Fig. 2(j) for rt = 221;) s Z Z s 4 4 (b) {4, 4}r,2: egc ⇔ (r, s) ∈ ([8, ∞)∩2 )×([4,r −4]∩2 ); (Fig. 2(e-f), rt =122, 102;) s Z Z s 3 (c) {4, 4}r,3: egc ⇔ (r, s) ∈ ([6, ∞) ∩ 6 ) × ([5,r − 3]\2 ); (Fig. 2(g) for rt =63;) s Z Z (d) {4, 4}r,t, t ∈ (4, ∞): egc ⇔ (r, t + s) ∈ (2 ) × (2 ). Proof. See Subsections 3.1, 3.2 and 3.3, Figures 2, 3 and 4 and Tables I, II and III for details of the proof. Observe that the restrictions in items 3(a) and 3(c) exclude the degenerate 1 3 cases of the 6-cycle {4, 4}6,1 and the complete bipartite graph {4, 4}6,3 = K3,3. Conjecture 5. The list in Theorem 4 covers all 34- 3222- and 24-graphs, in which the egc 34- 3222- and 24-graphs are those so indicated in items 1(b), 2 and 3.

3.1 The 4-cube as a twice-egc girth-4-regular graph Consider the three mutually orthogonal Latin squares [5] of order 4 contained in the following compound matrix, formed by the set of three MOLS(4):

1 2 4 7 0 111 222 333 444 3 243 134 421 312 (1) 5 324 413 142 231 6 432 341 214 123 where row and column headings stand for the following 4-tuples:

0=0000, 1=1000, 2=0100, 3=1100, 4=0010, 5=1010, 6=0110, 7=1110, (2) 0′ =0001, 1′ =1001, 2′ =0101, 3′ =1101, 4′ =0011, 5′ =1011, 6′ =0111, 7′ =1111.

Based on display (1), the top of Fig. 3 contains three copies of K4,4 properly colored in a mutually orthogonal fashion, where 1 is red, 2 is blue, 3 is green and 4 is hazel, as in Fig. 2(d-j). In the bottom of Fig. 3, three corresponding copies of the colored inverse images −1 φ (K4,4), where φ : Q4 → K4,4 is the canonical projection map, are obtained by identifying 0 the pairs of antipodal vertices of Q4 = {4, 4}4,4, these vertices denoted as in display (2). The leftmost copy of Q4 in Fig. 3 has color i attributed precisely to those edges parallel to th the i coordinate direction, for i =1, 2, 3, 4. This constitutes a 1-factorization F0 of Q4. On the other hand, he center and rightmost copies of Q4 in the ﬁgure determine 1-factorizations

7 j F1 and F2 of Q4 for which each girth cycle of Q4 intersects every composing 1-factor Fi of Fi, where i =1, 2 and j = 1 (red), 2 (blue), 3 (green), 4 (hazel). j For each edge e of Q4, we say that e has i-color j if e ∈ Fi . Then, the 0-color 4-cycles of Q4 have opposite edges with a common 0-color, with a total of two (nonadjacent) 0-colors per 4-cycle, say ℓ1,ℓ2 ∈{1, 2, 3, 4}, so one such 0-color 4-cycle can be expressed as (ℓ1,ℓ2,ℓ1,ℓ2). Moreover, the i-color 4-cycles of Q4 use all four colors 1,2,3,4, once each, for i =1, 2. There are 24 4-cycles in Q4, 6 of each of the 0-color 4-cycles expressed in the ﬁrst two columns of the following array, with two complementary 0-color 4-cycles per row. On the other hand, the third and fourth columns here contain respectively the 1-color and 2-color 4-cycles corresponding to the 0-color 4-cycles in the ﬁrst two columns:

(1212) (3434) (1234) (1243) (1313) (2424) (1324) (1342) (1414) (2323) (1423) (1432)

Figure 3: Factors of K4,4 and toroidal cutouts of Q4 with 0-, 1- and 2-color 4-cycles

3.2 Toroidal representation tables

Remark 6. The triple array in Table I below presents Fi (i =0, 1, 2) in schematic representa- 0 tions of Q4 = {4, 4}4,4, where ◦ stands for a vertex of Q4 and stands for an i-color 4-cycle. (This table is presented to establish a pattern for similar tables, like Table II below).

8 We say that F1 (resp. F2) has aւ12,34-zigzags and bց13,24-zigzags (resp. aց13,24-zigzags and bւ12,34-zigzags), meaning: (1) in F1, that: (a) any path obtained by walking left, down, left, down and so on, alternates either colors 1 and 2, or colors 3 and 4; (b) any path obtained by walking right, right, down, down and so on, alternates either colors 1 and 3, or colors 2 and 4; (2) in F2, that: (a) any path obtained by walking left, left, down, down and so on, either alternates colors 1 and 2, or colors 3 and 4; (b) any path obtained by walking right, down, right, down and so on, either alternates colors 1 and 3, or colors 2 and 4. By associating the oriented quadruple (1,3,2,4) (resp. (3,4,1,2)) of successive edge colors on the left-to-right or the downward (resp. the right-to-left or the downward) straight paths in F1 (resp. F2), situations that we indicate by “ց (1, 3, 2, 4)” (resp. “ւ (3, 4, 1, 2)”), a complete invariant for F1 (resp. F2) is obtained that we denote by combining between square brackets the just presented notations:

[aւ12,34, bց13,24, ց (1, 3, 2, 4)], (resp. [aց13,24, bւ12,34, ւ (3, 4, 1, 2)]).

This invariant distinguishes F1 and F2 from each other and is generalized for the toroidal graphs in Theorem 4, as we will see below in this subsection.

TABLE I

◦ 3 ◦ 2 ◦ 3 ◦ 2 ◦ ◦ 1 ◦ 3 ◦ 2 ◦ 4 ◦ ◦ 2 ◦ 1 ◦ 4 ◦ 3 ◦ 4 4 4 4 4 2 4 1 3 2 1 4 3 2 1 ◦ 3 ◦ 2 ◦ 3 ◦ 2 ◦ ◦ 3 ◦ 2 ◦ 4 ◦ 1 ◦ ◦ 3 ◦ 2 ◦ 1 ◦ 4 ◦ 1 1 1 1 1 4 1 3 2 4 2 1 4 3 2 ◦ 3 ◦ 2 ◦ 3 ◦ 2 ◦ ◦ 2 ◦ 4 ◦ 1 ◦ 3 ◦ ◦ 4 ◦ 3 ◦ 2 ◦ 1 ◦ 4 1 4 1 4 1 3 2 4 1 3 2 1 4 3 ◦ 3 ◦ 2 ◦ 3 ◦ 2 ◦ ◦ 4 ◦ 1 ◦ 3 ◦ 2 ◦ ◦ 1 ◦ 4 ◦ 3 ◦ 2 ◦ 1 1 1 1 1 3 2 4 1 3 4 3 2 1 4 ◦ 3 ◦ 2 ◦ 3 ◦ 2 ◦ ◦ 1 ◦ 3 ◦ 2 ◦ 4 ◦ ◦ 2 ◦ 1 ◦ 4 ◦ 3 ◦

A representation as in Table I may be used for the graphs in items 1(b) and 2 of The- orem 4. For example, the cases (r, t, s) = (20, 1, 5) and (r, t, s) = (28, 1, 5) in Fig. 2(h-i) are representable as in Table II below, where, instead of ◦ to represent each vertex, we set the vertex notation of Fig. 2(h-i). Here the four colors are indicated as in Fig. 2(d-j) and 5 Subsection 3.1. To distinguish the two cases in Table II, note that the 4-cycles of {4, 4}20,1 5 (resp. {4, 4}28,1) have the 2-factors by color pairs {1, 2} and {3, 4} descending in zigzag from right to left (resp. left to right), by alternate vector displacements (−1, 0) (resp. (1, 0)) for colors 1 and 3, and (0, −1) for colors 2 and 4. Generalizing and using the invariant notation of Remark 6, we can say that the egc graphs in item 2(b) of Theorem 4 are as follows:

5 1. {4, 4}6x+2,1 for x> 2 has invariant [aւ12,34, bց13,24, ց (2, 4, 1, 3)];

5 2. {4, 4}6x+4,1 for x> 2 has invariant [aւ13,24, bց12,34, ւ (2, 4, 3, 1)]. Observe that both in Remark 6 and here there is a distinguished oriented slanted arrow triple: either [ւ,ց , ց] or [ց,ւ , ւ]. Notice that the graphs in item 2 of Theorem 4 admit

9 both invariants. Setting this type of notation is simpler for the remaining toroidal graph cases in Theorem 4, exempliﬁed in Fig. 2, because the horizontal paths and vertical paths in those cases need a pair of alternate colors in each case, and the four colors in consideration are fully employed. The egc results above are condensed in the following statement, that take also into accounting the bipartite complement of the Heawood graph in Subsection 3.3.

TABLE II

00 2 01 4 02 1 03 3 04 2 05 00 2 01 1 02 3 03 4 04 2 05 1 06 3 07 1 3 2 4 1 3 1 3 4 2 1 3 4 2 05 4 06 1 07 3 08 2 09 4 10 07 4 08 2 09 1 10 3 11 4 12 2 13 1 14 3 2 4 1 3 2 2 1 3 4 2 1 3 4 10 1 11 3 12 2 13 4 14 1 15 14 3 15 4 16 2 17 1 18 3 19 4 20 2 21 2 4 1 3 2 4 4 2 1 3 4 2 1 3 15 3 16 2 17 4 18 1 19 3 00 21 1 22 3 23 4 24 2 25 1 26 3 27 4 00 4 1 2 3 4 1 3 4 2 1 3 4 2 1 00 2 01 4 02 1 03 3 04 2 05 00 2 01 1 02 3 03 4 04 2 05 1 06 3 07

3.3 The bipartite complement of the Heawood graph The bipartite complement H of the Heawood-graph, with vertex set V (H)={ij; i ∈{+, −}, j ∈ Z7}, is depicted on the upper left of Fig. 4; its edges {+j, −j}, {+j, −(j +2)}, {+j, −(j + 3)} and {+j, −(j + 4)}, for j ∈ Z7, will be denoted jj, j(j + 2), j(j + 3) and j(j + 4), respectively, where addition is taken mod 7. This yields the 28 edges of H as arcs from + to − vertices. They form 21 4-cycles ai, bi,ci (i ∈ Z7) expressed, by omitting the signs ±, as in Table III.

TABLE III

a0 = (00, 03, 33, 30), b0 = (00, 05, 52, 20), c0 = (02, 26, 63, 30), a1 = (11, 14, 44, 41), b1 = (11, 16, 63, 31), c1 = (13, 30, 04, 41), a2 = (22, 25, 55, 52), b2 = (22, 20, 04, 42), c2 = (24, 41, 15, 52), a3 = (33, 36, 66, 63), b3 = (33, 31, 15, 53), c3 = (35, 52, 26, 63), a4 = (44, 40, 00, 04), b4 = (44, 42, 26, 64), c4 = (46, 63, 30, 04), a5 = (55, 51, 11, 15), b5 = (55, 53, 30, 05), c5 = (50, 04, 41, 15), a6 = (66, 62, 22, 26), b6 = (66, 64, 41, 16), c6 = (61, 15, 52, 26).

This way, jj = aj ∩ aj+4 ∩ bj, j(j +2) = bj ∩ bj+2 ∩ cj, j(j +3) = aj ∩ cj ∩ cj+1 and j(j +4) = aj+4 ∩ bj+2 ∩ cj+1, ∀j ∈ Z7. We show there is no proper edge coloring of H that is tight on every 4-cycle. To prove this, we recur to the bipartite graph GA(H) whose parts V1 and V2 are respectively the 28 edges of H and the 21 4-cycles of H, with adjacency between an edge ij of H and a 4-cycle C of H whenever C passes through ij; GA(H) is represented in Fig. 4(b) with ai written as ai (i =1, 2, 3). A tight factorization of H would be equivalent to a 4-coloring of GA(H) that is monochromatic on each vertex of V1 but covering the four

10 colors at the edges incident to each vertex of V2. We begin by coloring the edges incident to vertices 00, 02, 03, 04 respectively with colors black, red, blue and green. This forces the coloring of the subgraph of GA(H) in the lower left of Fig. 4. By transferring this coloring to the representation of GA(H) on the right of Fig. 4, as shown, it is veriﬁed that vertex 15 on the bottom of the representation does not admit properly any of the 4 employed colors.

Figure 4: Bipartite complement H of the Heawood-graph and associated graph GA(H)

4 Prisms of girth-3-regular graphs

′ ′ ′ ′ Given a graph Γ , the prism graph P rism(Γ ) of Γ is the graph cartesian product K2Γ . 4 The remaining cases of (e1)(e2)(e3)(e4)-graphs with (e1)(e2)(e3)(e4) =6 1 , apart from those ′ ′ discussed in Section 3, are the prisms Γ = P rism(Γ ) of (e1)(e2)(e3)-graphs Γ . It is easy to see that there is no egc Γ if g(Γ′) is odd.

Conjecture 7. Graphs Γ with signatures 3221 and 433 are not egc.

Conjecture 7 is sustained by the exhaustive partial colorings of the prisms of the 24-vertex truncated octahedral graph ([11], 79–86) in Fig. 5(a-g), and the 6-vertex Thomsen graph (K3,3)[10] in Fig. 5(h), with the incidental obstructions indicated by a notiﬁcation ”No!” in each case. Such exhaustive partial colorings can be found similarly for example in the 120-vertex truncated-icosidodecahedral graph ([11], 97–99, or Wikipedia).

11 3 ′ ′ On the other hand, the 31 -graphs Γ = K2Γ with Γ being a (toroidal) quotient graph of the hexagonal tessellation [7, 8] (i.e. the tiling of the Euclidean plane of Schl¨aﬃ symbol {6, 3}) are egc 313-graphs, as illustrated in Fig. 5(i-j), namely for the prisms of the 24-vertex star graph ST4 (with twelve girth 6-cycles) [13] and a 16-vertex graph (with just eight girth 6-cycles). (This cannot be done with the Pappus graph, whose prism is drawn in Fig. 6(a); a fundamental region in this case (Fig. 6(e) with octodecimal vertex notation and proper face coloring) contains nine hexagonal tiles, not a multiple of 4, invalidating the proof of Theorem 8, below). So, we have the following.

2 3 3 Figure 5: Prisms of: truncated octahedron (32 1), K3,3 (4 3), ST4 and 16-vertex graph (31 )

Theorem 8. Let Γ′ be a toroidal graph given as quotient graph of the hexagonal tessellation with a number of hexagonal tiles divisible by 4 in any fundamental region of the tessellation. Then, the prism Γ of Γ′ is an egc 313-graph.

12 ′ Proof. Recall Γ = K2Γ . A tight factorization F = {F1 = Fblack, F2 = Fred, F3 = Fblue, F4 = Fgreen} of Γ is obtained for which each copy H of the hexagonal prism K2C6 in Γ corresponds to a 1-factor FH = Fi of F (i ∈ {1, 2, 3, 4}) that intersects H in a 1-factor Fi ∩ E(H) = FH ∩ E(H) of H comprising 6 edges. This establishes a color denomination for the copies H, as in Fig. 5(i-j), where such denomination (black, red, blue, green) is written in place in each case i, in each of the two instances of the ﬁgure. Each of the other three 1-factors (colors) contains four edges in each H (namely, two edges {(v0,u), (v1,u)} and ′ ′ ′′ ′′′ ′ {(v0,u ), (v1,u )} at distance 3 and two other edges {(v0,u ), (v0,u )} and {(v1,w), (v1,w )}, also at distance 3), subtotaling 3×4 = 12 edges, apart from the six edges of FH ∩H, totaling the 18 edges of H. Clearly, all these partial 1-factorizations integrate into F .

Figure 6: Pappus, Desargues, Nauru and Dyck prisms

Notice that the argument in the previous paragraph only holds if the imposed condition that the hexagonal tessellation in the statement has a number of hexagonal tiles divisible by 4 in any fundamental region. This is due to the fact that the copies of H in the previous paragraph must have the same number of hexagonal prisms in each of the four colors, namely black, red, blue and green. The Pappus and Desargues graphs are shown not to be egc via obstructions formed by pairs of forced “long” parallel red edges in Fig. 6(a-b). However, in Fig. 6(c-d) the Nauru and

13 Dyck graphs, respectively, are shown to be egc by means of corresponding tight factorizations in their representations.

TABLE IV 3¯ 34123¯ 4¯ 34213¯ 4¯ 43214¯ 3¯ 43124¯ ◦ 4 ◦ 1 ◦ 2 ◦ 3 ◦ 4 ◦ 1 ◦ 2 ◦ 1 ◦ 3 ◦ 4 ◦ 2 ◦ 1 ◦ 3 ◦ 2 ◦ 1 ◦ 4 ◦ 3 ◦ 2 ◦ 1 ◦ 2 ◦ 4 ◦ 3 ◦ 1 ◦ 2 1 2 3 4 1 2 3 4 2 1 3 4 2 1 4 3 2 1 4 3 1 2 4 3 ◦ 3 ◦ 4 ◦ 1 ◦ 2 ◦ 3 ◦ 4 ◦ 1 ◦ 3 ◦ 4 ◦ 2 ◦ 1 ◦ 3 ◦ 4 ◦ 3 ◦ 2 ◦ 1 ◦ 4 ◦ 3 ◦ 2 ◦ 4 ◦ 3 ◦ 1 ◦ 2 ◦ 4 2¯ 12341¯ 2¯ 21342¯ 1¯ 21432¯ 1¯ 12431¯

The discussion above in relation to Fig. 5-6 and Theorem 8 yields a setting to get or disproof that a prism graph Γ = P rism(Γ′) is an egc graph when g(Γ′) = 6. This depends on the existence of a predominant edge color in each copy of K2H, for each girth 6-cycle H of Γ, as in the proof of Theorem 8, where the edge-color distribution of each H is (6, 4, 4, 4), i.e. the predominant color assigned to six of the edges. In case g(Γ′) = 8, let us consider Γ′ to be the Tutte 8-cage (or Tutte-Coxeter or Cremona- Richmond graph). Fig. 7(a-c) shows why its 313-graph prism is not egc, with 8-cycle prisms in Γ having edge-color distribution (6,6,6,6), presenting partial edge-colorings in Γ, with copies of K2C8 edge-colored with the said distribution (6,6,6,6) and notification “No!” if an obstruction to edge-coloring continuation appears. On the other hand, Table IV uses the notation of Table I in representing a (6, 6, 6, 6)- 3 distributed coloring of the union U of almost four (namely 3 4 ) contiguous 8-cycle prisms Θi (i = 1, 2, 3, 4) and the resulting forced colors for the departing edges away from U. In Table IV, the middle row sequence, call it Υ, (obtained by disregarding the symbols “”, or replacing them by commas) represents the subsequences of colors of the edges {(0,u), (1,u)} 2 2 in the Θi’s, namely the corresponding subsequences Υ1 = (1, 2, 3, 4) , Υ2 = (3, 4, 2, 1) , 2 2 Υ3 = (2, 1, 4, 3) and Υ4 = (4, 3, 1, 2) . Here, the last two terms of each Υi coincide (i.e. are shared) with the first two terms of its subsequent Υi+1, where the last 6-term subsequence is completed to Υ4 by adding the first two terms of Υ1, soΥ1 may be considered as the next 3 Υi after Υ4 (and explaining the fraction 3 4 mentioned above). This suggest that Υ can be concatenated with itself a number ℓ of times in order to close a 24 × ℓ-cycle of colors for the edges ({0,u), (1,u)} of a Hamilton-cycle (of Γ′) prism H which may be completed to an egc graph Γ (see Fig. 7(d)) by means of the following considerations. (Fig. 7(e) needed in Section 6 for Fig. 12). On the top and bottom rows of Table IV, the colors 1, 2, 3 and 4 with a bar on top are the colors of the “long” edges in the four represented 8-cycle prisms Θi in Fig. 7(d) that close the two corresponding 8-cycles in each Θi. The remaining (non-barred) colors suggest that the corresponding edges form external 4-cycles that may be joined with H in order to form a Γ as desired. The “even longer” edges of these external 4-cycles (edges not of the form (0,u)(1,u)) must be set to form (with two edges of the form {(0,u), (1,u)}) new 4-cycles and can in fact be selected in order to form the desired Γ by taking the number of concatenated copies of U to be ℓ = 4, so that |V (Γ)| = 192. For example, the two columns in Table IV whose transpose rows are “4 ◦ 3 ◦ 2” and “4 ◦1◦2”, namely the 3rd leftmost and 7th rightmost -free columns, integrate one such 4-cycle. In fact, the leftmost 3rd, 4th, 5th

14 and 6th columns are paired this way with the rightmost 7th, 8th, 9th and 10th columns, but because of the selection above the last three pairs must be paired with similar columns in the 2nd, 3rd and 4th version of Table IV (for indices k ∈ {2, 3, 4= ℓ} of copies Uk of U, if we agree that the leftmost 3rd and rightmost 7th columns selected above are both for k =1 and U = Uk = U1). The same treatment can be set from the leftmost 9th, 10th, 11th and 12th respectively to the rightmost 1st, 2nd, 3th and 4th columns, which also correspond in pairs that form again “long” 4-cycles.

Figure 7: Tutte 8-cage 313-graph prism, 96-vertex cubic Γ′ and Petersen graph

Theorem 9. While the Tutte 8-cage is a 313-graph which is not egc, we have that for each integer k ≥ 4 there exists an egc 313-graph Γ with 192 × k edges as a prism of a Hamiltonian cubic graph Γ′ on 96 × k vertices based on g contiguous copies of the edge-colored subgraph represented in Table IV.

Proof. The argument previous to the statement can be completed for the case k = 1. Clearly, by concatenating any multiple of g times the graph represented in Table IV, one arrives at eﬀectively extending the construction.

It remains to consider prisms of (e1)(e2)(e3)-graphs whose girths are even numbers larger

15 than 8. These include the 90-vertex Foster graph and the126-vertex Tutte’s 12-cage [3]. We pose the following.

Conjecture 10. No prism of an (e1)(e2)(e3)-graph whose girth is larger than 8 is egc.

5 Egc girth-4-regular graphs (continuation)

Figure 8: P(W (5, 2)) and tight factorization of P(W (5, 2))

The still remaining cases of (e1) ··· (eg)-graphs for g = 4 not treated above have (e1) = (e2)= ··· =(eg) = 1. In [23], these graphs are said to be girth-tight. A graph construction 4 [23, 28] for (e1)(e2)(e3)(e4)=1 used below in the present work is as follows. Let Γ be 4-regular and let C be a partition of E(Γ) into cycles. The pair (Γ, C) is said to be a cycle decomposition of Γ. Two edges of Γ are called opposite at vertex v if they are both incident to v and belong to the same element of C. The partial line graph P(Γ, C) of a cycle decomposition (Γ, C) is the graph whose vertices are the edges of Γ, with any two such edges adjacent as vertices of P(Γ, C) if they share a vertex of Γ and are not opposite at that vertex. Let (Γ, C) be a cycle decomposition. A cycle C in Γ is C-alternating if not two consecutive edges of C belong to the same element of C. In our case, Lemma 4.10 [23] says that P(Γ, C) is girth-tight if and only if (Γ, C) contains neither C-alternating cycles nor triangles except those contained in C. Remark 11. As initial examples of partial line graphs, consider the wreath graph W (n, 2) = Cn[K2] (n > 4), where Cn is a cycle (v0, v1,...,vn−1). Consider the partition C of W (n, 2) into the 4-cycles ((vi, 0), (vi+1, 0), (vi, 1), (vi+1, 1)), where i ∈ Zn. These cycles form a decomposition (W (n, 2), C), which determines the partial line graph P(W (n, 2), C). It can be seen that if n is: (a) odd, then P(W (n, 2), C) is not egc; see Fig. 8(a), showing an edge partition of P(W (5, 2), C) into two green 5-cycles, one blue 10-cycle and one hazel 20-cycle; (b) even,

16 then P(W (n, 2), C) is egc, as in Fig. 8(b), where P(W (6, 2), C) is represented showing such a tight factorization via edge colors red, blue, green and hazel. This establishes the following.

Figure 9: Tight factorizations of (a) P(Br(4, 5;2)) and (b) P(MBr(4, 8;3))

Theorem 12. Let 4 < n ∈ Z. Then, P(W (n, 2), C) is egc if and only if n ≡ 0 (mod 2).

A particular case of a cycle decomposition is that of a linking-ring structure [28], that works just for two colors, say red and green. This structure applies for us in the following paragraphs only for n even; (if n is odd then more than two colors would be needed in order to distinguish adjacent cycles of the decomposition (W (n, 2), C)). It is deﬁned as follows. An isomorphism between two cycle decompositions (Γ1, C1) and (Γ2, C2) is an isomorphism ξ :Γ1 → Γ2 such that ξ(C1)= C2. An isomorphism ξ from a cycle decomposition to itself is

17 an automorphism, written ξ ∈ Aut(Γ, C). A cycle decomposition (Γ, C) is ﬂexible if for every vertex v and each edge e incident to v there is ξ ∈ Aut(Γ, C) such that: (I) ξ ﬁxes each vertex of the cycle in C containing e and (II) ξ interchanges the two other neighbors of v; the edges joining v to those neighbors are in some other cycle of C. A cycle decomposition (Γ, C) is bipartite if C can be partitioned into two subsets G (green) and R (red) so that each vertex of Γ is in one cycle from G and one from R.

TABLE V

0 0 1 3 0 1 2 0 0 2 1 1 0 3 2 2 (01[a]11[b]02[c]01[d]) (12[a]22[b]14[c]12[d]) (21[a]31[b]22[c]21[d]) (32[a]02[b]34[c]32[d]) 1 0 2 3 1 1 3 0 1 2 2 1 1 3 3 2 (02[a]12[b]03[c]02[d]) (13[a]23[b]10[c]13[d]) (22[a]32[b]23[c]22[d]) (33[a]03[b]30[c]33[d]) 2 0 3 3 2 1 4 0 2 2 3 1 2 3 4 2 (03[a]13[b]04[c]03[d]) (14[a]24[b]11[c]14[d]) (23[a]33[b]24[c]23[d]) (34[a]04[b]31[c]34[d]) 3 0 4 3 3 1 0 0 3 2 4 1 3 3 0 2 (04[a]14[b]00[c]04[d]) (10[a]20[b]12[c]10[d]) (24[a]34[b]20[c]24[d]) (30[a]00[b]32[c]30[d]) 4 0 0 3 4 1 1 0 4 2 0 1 0 3 1 2 (00[a]10[b]01[c]00[d]) (11[a]21[b]13[c]11[d]) (20[a]30[b]21[c]20[d]) (31[a]01[b]33[c]31[d])

The largest subgroup of Aut(Γ, C) preserving each of the sets G and R is denoted Aut+(Γ, C), taken as the color-preserving group of (Γ, C). In a bipartite cycle decomposition, an element of Aut(Γ, C) either interchanges G and R or preserves each of G and R set-wise, so it is contained in Aut+(Γ, C). This shows that the index of Aut+(Γ, C) in Aut(Γ, C) is at most 2. If this index is 2, then we say that (Γ, C) is self-dual; this happens if and only if there is σ ∈ Aut(Γ, C) such that Gσ = R and Rσ = G. In [23], a cycle decomposition (Γ, C) is said to be a linking-ring (LR) structure if it is (i) bipartite, (ii) flexible and (iii) Aut+(Γ, C) acts transitively on V (Γ). However, there are tight factorizations of graphs P(Γ, P) obtained by relaxing in that definition condition (iii). So we will say that a cycle decomposition (Γ, P) is a relaxed LR structure if it satisfies just conditions (i) and (ii). Remark 13. With the aim of yielding semisymmetric graphs from LR structures, [28] defines: (a) the barrel Br(k, n; r), where 4 ≤ k ≡ 0 mod 2, n ≥ 5, r2 ≡ ±1 mod n, r 6≡ ±1 mod n Z Z n and 0 ≤ r < 2 , as the graph with vertex set g × n and (i, j) red-adjacent to (i ± 1, j) and green-adjacent to (i, j ± ri); (b) the mutant barrel MBr(k, n; r), where 2 ≤ k ≡ n ≡ 0 mod 2, n ≥ 6, r2 ≡ ±1 mod n and r 6≡ ±1 mod n, as the graph with vertex set Zg × Zn and (i, j) red-adjacent to (i +1, j) n for 0

18 TABLE VI

0 0 1 2 0 0 2 1 0 2 3 1 (01[a]11[b]02[c]01[d]) (12[c]12[d]14[a]22[b]) (23[a]03[b]26[c]23[d]) 1 0 2 2 1 0 3 1 1 2 4 1 (02[a]12[b]03[c]02[d]) (13[c]13[d]15[a]23[b]) (24[a]04[b]20[c]24[d]) 2 0 3 2 2 0 4 1 2 2 5 1 (03[a]13[b]04[c]03[d]) (14[c]14[d]16[a]24[b]) (25[a]05[b]21[c]25[d]) 3 0 4 2 3 0 5 1 3 2 6 1 (04[a]14[b]05[c]04[d]) (15[c]15[d]10[a]25[b]) (26[a]06[b]22[c]26[d]) 4 0 5 2 4 0 6 1 4 2 0 1 (05[a]15[b]06[c]05[d]) (16[c]16[d]11[a]26[b]) (20[a]00[b]23[c]20[d]) 5 0 6 2 5 0 0 1 5 2 1 1 (06[a]16[b]00[c]06[d]) (10[c]10[d]12[a]20[b]) (21[a]01[b]24[c]21[d]) 6 0 0 2 6 0 1 1 6 2 2 1 (00[a]10[b]01[c]00[d]) (11[c]11[d]13[a]21[b]) (22[a]02[b]25[c]22[d])

0 0 1 2 1 0 2 1 2 2 3 1 (01[a]11[b]05[c]05[d]) (12[c]12[d]10[a]20[b]) (23[a]03[b]21[c]21[d]) 1 0 5 2 2 0 0 1 3 2 1 1 (05[a]15[b]02[c]02[d]) (10[c]10[d]13[a]23[b]) (21[a]01[b]24[c]24[d]) 5 0 2 2 0 0 3 1 1 2 4 1 (02[a]12[b]04[c]04[d]) (13[c]13[d]15[a]25[b]) (24[a]04[b]20[c]20[d]) 2 0 4 2 3 0 5 1 4 2 0 1 (04[a]14[b]03[c]03[d]) (15[c]15[d]14[a]24[b]) (20[a]00[b]25[c]25[d]) 4 0 3 2 5 0 4 1 0 2 5 1 (03[a]13[b]06[c]06[d]) (14[c]14[d]16[a]26[b]) (25[a]05[b]26[c]26[d]) 3 0 6 2 4 0 6 1 5 2 6 1 (06[a]16[b]00[c]00[d]) (16[c]16[d]11[a]21[b]) (26[a]06[b]22[c]22[d]) 6 0 0 2 6 0 1 1 6 2 2 1 (00[a]10[b]01[c]01[d]) (11[c]11[d]12[a]22[b]) (22[a]02[b]23[c]23[d])

0 0 1 2 0 0 3 1 0 2 2 1 (01[a]11[b]06[c]01[d]) (13[c]13[d]11[a]21[b]) (22[a]02[b]21[c]21[d]) 1 0 6 2 3 0 1 1 2 2 1 1 (06[a]16[b]00[c]06[d]) (11[c]11[d]15[a]25[b]) (21[a]01[b]24[c]24[d]) 6 0 0 2 1 0 5 1 1 2 4 1 (00[a]10[b]01[c]00[d]) (15[c]15[d]10[a]20[b]) (24[a]04[b]20[c]20[d]) 2 0 3 2 5 0 0 1 4 2 0 1 (03[a]13[b]04[c]03[d]) (10[c]10[d]13[a]23[b]) (20[a]00[b]22[c]22[d]) 3 0 4 2 2 0 4 1 3 2 5 1 (04[a]14[b]05[c]04[d]) (14[c]14[d]16[a]26[b]) (25[a]05[b]26[c]26[d]) 4 0 5 2 4 0 6 1 5 2 6 1 (05[a]15[b]02[c]05[d]) (16[c]16[d]12[a]22[b]) (26[a]06[b]23[c]23[d]) 5 0 2 2 6 0 2 1 6 2 3 1 (02[a]12[b]03[c]02[d]) (12[c]12[d]14[a]24[b]) (23[a]02[b]25[c]25[d]) Note that thick edges of colors orange and black form cycles zigzagging between: (A) the vertices of each vertical green cycle in the ﬁgure (excluding the rightmost green cycle) and (B) their adjacent red vertices to their immediate right. Also, note that thick blue and hazel edges form cycles zigzagging between: (C) the red vertices and (D) the vertices of the next vertical green cycle to their right. The girth is realized by tight 4-cycles (i.e. using the four colors) with a pair of edges (blue and hazel) to the left of each vertical green cycle and another pair of edges (black and red) to the corresponding right. This is always attainable, because similar bicolored cycles can always we obtained, generating the desired tight factorizations. A code representation of the tight factorization in Fig. 9(a) is given in Table V, where j+1 each green edge {ij, i(j + 1)} contains a green vertex denoted ij , each red edge {ij, hj} h contains a red vertex denoted i j, and the color of an edge between a green vertex and a red vertex is indicated between brackets: [a] for orange, [b] for black, [c] for hazel and [d] for blue. Remark 14. Generalizing Remark 13 to determine other egc girth-tight graphs, we consider n n n n a 2-factorization F = {F1 , F2 ,...,Fk−1} of the complete graph Kn, for odd n =2k +1 > 6 n and use it to deﬁne the barrel Br(k, F ), with (a) Zg × Zn as vertex set and (b) edges forming precisely red cycles ((0, i), (1, i),..., (k − 1, i)), where i ∈ Zn, and green subgraphs n Z {j} × Fj , where j ∈ g.

19 Figure 10: Egc P(Br(3, F )) for the three 2-factorizations F of K7

20 Fig. 10 contains representations of P(Br(3, F 7)) for three distinct 2-factorizations F 7 of K7, with tight factorizations represented as in Fig. 9, but here green cycles are represented so that each vertex (i, 0) = i0 appears just once (not twice, as in Fig. 9(a-b)). In the three cases, 7 7 7 F1 , F2 and F3 , green edges are traced thick, thin and dashed, respectively. To the left of these representations, the corresponding green subgraphs are shown. Code representations of these three tight factorizations can be found in Table VI, following the conventions of Table V. (A diﬀerent 1-factorization of K7 that may be used with the same purpose is for example {(0, 1, 2, 3, 4, 5, 6), (0, 3, 5, 1, 6, 2, 4), (0, 2, 5)(1, 3, 6, 4)).

P 8 8 8 Figure 11: Tight factorization of (MBr(3, {F1 , F2 , F3 })), based on 2-factors of K8

TABLE VII

0 0 1 3 0 1 2 0 0 2 3 1 0 3 4 2 (01[a]11[b]02[c]01[d]) (12[a]22[b]14[c]12[d]) (23[a]33[b]26[c]23[d]) (34[a]04[b]38[c]34[d]) 1 0 2 3 1 1 3 0 1 2 4 1 1 3 5 2 (02[a]12[b]03[c]02[d]) (13[a]23[b]15[c]13[d]) (24[a]34[b]27[c]24[d]) (35[a]05[b]30[c]35[d]) 2 0 3 3 2 1 4 0 2 2 5 1 2 3 6 2 (03[a]13[b]04[c]03[d]) (14[a]24[b]16[c]14[d]) (25[a]35[b]28[c]25[d]) (36[a]06[b]31[c]36[d]) 3 0 4 3 3 1 5 0 3 2 6 1 3 3 7 2 (04[a]14[b]05[c]04[d]) (15[a]25[b]17[c]15[d]) (26[a]36[b]20[c]26[d]) (37[a]07[b]32[c]37[d]) 4 0 5 3 4 1 6 0 4 2 7 1 0 3 8 2 (05[a]15[b]06[c]05[d]) (16[a]26[b]18[c]16[d]) (27[a]37[b]21[c]27[d]) (38[a]08[b]33[c]38[d]) 5 0 6 3 1 1 7 0 5 2 8 1 1 3 0 2 (06[a]16[b]07[c]06[d]) (18[a]27[b]10[c]17[d]) (28[a]38[b]22[c]28[d]) (30[a]00[b]34[c]30[d]) 6 0 7 3 2 1 8 0 6 2 0 1 2 3 1 2 (07[a]17[b]08[c]07[d]) (10[a]28[b]11[c]18[d]) (20[a]30[b]23[c]20[d]) (31[a]01[b]35[c]31[d]) 7 0 8 3 3 1 0 0 7 2 1 1 3 3 2 2 (08[a]18[b]00[c]08[d]) (11[a]20[b]12[c]10[d]) (21[a]31[b]24[c]21[d]) (32[a]02[b]36[c]32[d]) 8 0 0 3 4 1 1 0 8 2 2 1 0 3 3 2 (00[a]10[b]01[c]00[d]) (12[a]21[b]13[c]11[d]) (22[a]32[b]25[c]22[d]) (31[a]03[b]37[c]33[d])

In the same way, by considering the 2-factorization given in K9 seen as the Cayley graph 9 9 9 9 9 C9(1, 2, 3, 4) with F formed by the 2-factors F1 , F2 , F3 , F4 generated by the respective colors 9 9 9 9 1, 2, 3, 4, namely Hamilton cycles F1 , F2 , F4 but F3 =3K3, we get a tight factorization of P(4, F 9). This is encoded in Table VII in similar fashion to that of Tables IV and V.

21 n n n n A similar generalization takes a 2-factorization F = {F1 , F2 ,...,Fk−2} of Kn −{i(i + k); i =0,...,k − 1}, where n =2k is even, and uses the 1-factor {i(i + k); i =0,...,k − 1} to get a generalized mutant graph MBr(k − 1, F n) in a likewise fashion to that of item (b) n in Remark 13 but modified now via F , with Zk−1 × Zn as vertex set. For example, Fig. P 8 8 8 8 8 11 represents a tight factorization in (MBr(3, F )), where F = {F1 , F2 , F3 }, represented 8 8 on the upper left of the figure, is such a 2-factorization, with F1 and F3 as in Fig. 10, and 8 F2 = (0, 2, 4, 6)(1, 3, 5, 7), via corresponding thick, thin and dashed, green edge tracing. On the lower left, a representation of the red-green graph MBr(3, F 8) is found. We can further extend these notions of barrel and mutant barrel by taking a cycle Gn = n n n n n n (G1 ,G2 ,...,Gℓ ) of copies of the 2-factors of F , where Gi ∈ F but with no two contiguous n n n n Gi and Gi+1 mod n being the same element of F . Here, ℓ ≥ 3. This defines barrel Br(ℓ,G ) or mutant barrel MBr(ℓ,Gn) (n even in this case) and establishes the following. Theorem 15. The barrels and mutant barrels finally obtained in Remark 14 produce corresponding egc graphs P(Br(ℓ,Gn)) and P(MBr(ℓ,Gn)).

6 Egc girth-5-regular graphs

An egc 5302-graph is presented by means of a construction that generalizes the barrel con- structions used in Section 5, as follows. See Fig. 12, where 14 vertical copies of the Petersen graph P et are presented in parallel at equal distances from left to right and numbered from Z j j j j 0 to 13 in 14. The vertices of the j-th copy of P et, call it P et , are denoted v1, v1,...,v10 from top to bottom and are joined horizontally by cycles of the Cayley graph of Z14 with generator set {1, 3, 5}, namely the cycles

0 1 2 3 4 5 6 7 8 9 10 11 12 13 (vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi ), for i =1, 5, 7, 9; 0 3 6 9 12 1 4 7 10 13 2 5 8 11 (vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi ), for i =2, 3, 4; 0 5 10 1 6 11 2 7 12 3 8 13 4 9 (vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi , vi ), for i =6, 8, 10.

Figure 12: An egc 5302-graph on 140 vertices via 14 copies of the Petersen graph

22 Theorem 16. There exists an egc 5302-graph of order 140 × k, for every 0 < k ∈ Z, representing all color-cycle permutations, 14k times each.

Proof. We assert that an egc 5302-graph as in the statement contains either 14k disjoint copies of the Petersen graph or, if 1 < k ≡ 0 mod 2, 7k disjoint copies of the dodecahedral graph. First, we consider the case represented in. Fig. 12. Notice the 6-cycle j j j j j j j j j j j j (v5, v6, v7, v8, v9, v10, v11) in P et , with its three pairs of opposite vertices {v5, v8}, {v6, v9}, j j j j j j Z {v7, v10} joined respectively to the neighbors v2, v3, v4 of the top vertex v1, for j ∈ 14. A representation of the common proper coloring of the graphs P etj is in Fig. 7(e), where the j Z vertices vi are simply denoted i, for i =1,..., 10 and j ∈ 14. This ﬁgure shows the twelve possible 5-cycles of P et and corresponding color cycles, where red, black, blue, hazel and green are indicated respectively 1,2,3,4 and 5. This gives the following 1-to-1 correspondence from the 5-cycles of P et to their color 5-cycles:

(1, 2, 5, 10, 4) → (1, 2, 3, 4, 5); (3, 6, 7, 8, 9) → (4, 1, 3, 5, 2); (4, 1, 2, 8, 7) → (5, 1, 4, 3, 2); (10, 9, 3, 6, 5) → (1, 2, 4, 5, 3); (1, 2, 5, 6, 3) → (1, 2, 5, 4, 2); (4, 7, 8, 9, 10) → (2, 3, 5, 1, 4); (2, 5, 10, 9, 8) → (2, 3, 1, 5, 4); (1, 3, 6, 7, 4) → (3, 4, 1, 2, 5); (5, 10, 4, 7, 6) → (3, 4, 2, 1, 5); (2, 8, 9, 3, 1) → (4, 5, 2, 3, 1); (10, 4, 1, 3, 9) → (4, 5, 3, 2, 1); (5, 6, 7, 8, 2) → (5, 1, 3, 4, 2). In fact, there are exactly 12 possible color 5-cycles and they are the targets of the 1-to-1 correspondence. They are obtained from the 5!=120 permutations on 5 elements as the 12 orbits of the dihedral group D10 generated both by translations mod 5 and by reﬂections on the 5-tuples on {1, 2, 3, 4, 5}. This insures the statement for k = 1, since the 12 vertical copies P etj of P et are the only source of the color cycles. The extension of this for any 0 < k ∈ Z is immediate. Also, the dodecahedral graph Dod, that has an homomorphism onto P et [1] given by identifying antipodal vertices, can be used instead of P et to produce graphs as those obtained above but based on Dod, which insures the statement. A barrel-type construction as that of the 140-vertex 5302-graph above can be used to combine P et (resp. Dod) and the Cayley graph of Z7 with generator set {1, 2, 4} and obtain a 70-vertex (resp. 140-vertex) 5302-graph with other 5-cycles besides the vertical ones. In the P et case, the horizontal orbits, of lenght 7, would be taken to be:

0 1 2 3 4 5 6 (vi , vi , vi , vi , vi , vi , vi ) for i =1, 4, 6, 8; 0 2 4 6 1 3 5 (vi , vi , vi , vi , vi , vi , vi ) for i =2, 3, 4; 0 4 1 5 2 6 3 (vi , vi , vi , vi , vi , vi , vi ) for i =5, 7, 9. Theorem 17. Let TI be the 60-vertex truncated-icosahedral graph. Then, there is an egc 104-graph on 840.k vertices and a non-egc 104-graph on 420.k vertices, for each integer k > 0.

Proof. By means of a barrel-type construction as in Fig. 12, one can combine 14k copies of 4 TI and the Cayley graph of Z14 with generator set {1, 3, 5} to get an egc 10 -graph on 840k vertices. It also can be seen that a 104-graph exists on 420 vertices by combining 7 copies of TI and the Cayley graph of Z7 with generator set {1, 2, 4}.

23 Theorem 18. The four 30-vertex (5,5)-cages [21] and the 36-vertex Sylvester graph [3] are 55-graphs, but none of them is egc.

Figure 13: Foster cage, Meringer, Wong, Robertson-Wenger and Sylvester graphs (see text)

Proof. Clearly, the ﬁve graphs in the statement are 55-graphs. Fig. 13 depicts the Foster cage Fc and Meringer graph Mer on top in gray and red, with a common distinguished subgraph on red edges again depicted to their right as the union of the 5-cycles C0 =(abcde), C1 = (aefgh), C2 = (abcgh), C3 = (cdefg), C4 = (bcdij), C5 = (hidea), C6 = (hijba), C7 =(cghid), C8 =(akmde) and C9 =(ablne). The ﬁrst eight of these ten 5-cycles can be

24 tightly colored in a unique way up to color permutations, as suggested in the figure, but it is not possible to color tightly the 5-cycle C8. Thus, Fc and Mer are not egc. The middle two gray-and-red 5-cages in Fig. 13 are the Wong graph Wo and and Rober- son-Wenger graph R − W , each with a common red subgraph again depicted to their right ′ ′ ′ as the union of the 5-cycles C0,C1,C2,C3,C4, C5 = ((bcdmn), C6 =(bnkha), C7 =(kijbn), ′ ′ C8 =(kidmn) and C9 =(defol). The first six of these ten 5-cycles can be tightly colored in a unique way up to color permutations, as suggested in the figure, but it is not possible to color tightly the three 5-cycles containing the still non-used vertex k. Thus, Wo and R − W are not egc. The bottom graph in Fig. 13 is the Sylvester graph Syl, with a red subgraph again depicted to its right as the union of the 5-cycles C0,C1,C4,C8,C9, C10 = (abcpo), C11 = (cderq), C12 = (aefso), C13 = (eahtr), C14 = (aksvo) and C15 = (enutr). Coloring tightly the 5-cycles C0,C8,C9,C10 and C11 with color(ab)=color(ef) and color(ah)=color(de) results in the impossibility of continuing coloring tightly the 5-cycle C4, as can be seen to the right of Syl in Figure 13. Otherwise, a forced tight coloring of C0,...,C12, C14 and C15 is obtained, depicted further to the right, but that leaves C13 obstructing a tight coloring continuation. Thus, Syl is not egc. We mention that the 70.k-vertex graph based on P et, the 140.k-vertex graph based on Dod and the 420.k-vertex graph based on TI are 55-graphs, where 0

Conjecture 19. The smallest order of an (e1)(e2)(e3)(e4)(e5)-graph is 140 and this happens 3 2 for (e1)(e2)(e3)(e4)(e5)=5 0 .

7 Are there egc girth-6-regular graphs?

The Hamming code H7 in the 7-cube graph Q7 is generated by the binary matrix

1101000 0110100 0011010

Thus, H7 is a set of 7-tuples whose supports form an associated set given by the union of: the empty set, the Fano plane vertex set F = {124, 235, 346, 450, 561, 602, 013} (i.e., the Steiner triple system S(2, 3, 7) presented cyclically) and the complements of the eight resulting supports. Note that H7 is the kernel of a parity check matrix, so Q7 − H7 was referred in [12]asa Hamming shell, and later [20] called it the Dejter graph. Now, Q7 −H7 is a 6-regular graph whose girth is 6 [2], namely the disjoint union of 112 6-holes, i.e. induced (or chordless) 6-cycles. Thus, it makes sense trying to ﬁnd 1-factorizations of Q7 − H7. The antipodal quotient of Q7 − H7, call it AQ7, is the disjoint union of 56 6-cycles; see Table VIII, where vertices are pairs ab, with a as an antipodal Hamming-code class at distance one from ab, and b as the edge direction from ab to a. Those a denote as follows: 7 for the null 7-tuple, and 0,1,2,3,4,5,6 respectively for 124,235,346,450,561,602,013. Instead of separating commas, a subindex between two vertices in a 6-cycle in the table represents

25 the direction separating them. Superindices a, b, c, d, e, f deﬁne a 6-color assignment onto E(AQ7), 1-to-1 onto the edges of each 6-cycle in the table. Up to symmetry, this assignment is forced on AQ7. Colors d, b, f in the 2nd, 3rd and 4th 6-cycle columns, respectively, yield adjacent edges on a common color, and so are obstructions to a proper coloring.

TABLE VIII

b c d e f a b f a d c e f d e b a c d b c f e a (714021742014721 042) (314221 342214321242) (621 442614421642414) (542114521142 514121) b c d e f a b f a d c e f d e b a c d b c f e a (725132753125732 153) (425332 453325432353) (032 553025532053525) (653225632253 625232) b c d e f a b f a d c e f d e b a c d b c f e a (736243764236743 264) (536443 564436543464) (143 664136643164636) (064336043364 036343) b c d e f a b f a d c e f d e b a c d b c f e a (740354705340754 305) (640554 605540654505) (254 005240054205040) (105440154405 140454) b c d e f a b f a d c e f d e b a c d b c f e a (751465716451765 416) (051665 016651065616) (365 116351165316151) (216551265516 251565) b c d e f a b f a d c e f d e b a c d b c f e a (762506720562706 520) (162006 120062106020) (406 220462206420262) (320662306620 362606) b c d e f a b f a d c e f d e b a c d b c f e a (703610731603710 631) (203110 231103210131) (510 331503310531303) (431003410031 403010)

Theorem 20. Q7 − H7 does not have a tight factorization.

Proof. The color assignment above can be transformed in the inverse image of AQ7 as in the following image of a modiﬁcation of the ﬁrst row of Table VIII, in which each vertex ij is kept as it is if i = 7 and is replaced by the vertex 0(j − i) mod 7 if i =6 7:

b c d e f a b f a d c e f d e b a c d b c f e a (71a02d74c 01f 72e 04b )(05a00e 01b 06c 06d02f )(03e 00c 02f 05a05b 04d)(06c 00a04d03e 03f 01b ) where the subindices of the ﬁrst row are replaced by adequate colors for the forward edges from even-to-odd weight vertices of Q7 − H7 while the corresponding superindex colors are kept for the remaining edges of Q7 − H7. This color assignment resolves the obstruction above, but it fails around the vertices ij with j = i, namely in the second entry of the 2nd, 3d and 4th 6-cycle columns. This insures that Q7 − H7 is not egc.

Q7 − H7 decomposes into two isomorphic girth-10 cubic Ljubljana graphs [6] via an edge partition [2], and the inverse image of each edge of AQ7 in it is formed by an edge in each such Ljubljana graph. It seems that this is the only i-factorization of Q7 − H7, (0 < i ∈ Z).

8 Hamilton cycles and hamiltonian decomposabilty

Some feasible applications of egc graphs happen when the unions of pairs of composing 1-factors constitute Hamilton cycles, possibly attaining hamiltonian decomposability in the case of even-degree of such graphs. This oﬀers a potential beneﬁt to the applications drawn in Section 1, if an optimization/decision-making problem requires alternate inspections covering all nodes of the involved system, when the alternancy of two colors is required. In Fig. 1, cases (e-h) and their triangle-replaced graphs (m-p) have the unions of any two of their 1-factors forming a Hamilton cycle, while cases (i), (q) and (v) have those unions as disjoint pairs of two cycles of equal length. In particular, the 3-cube graph Q3 that admits just two tight factorizations, has one of them creating Hamilton cycles (case (h)) and the

26 other one with the second said property (case (i)). The triangle-replaced graph, ∇(Q3), has corresponding tight factorizations in cases (p-q) with similar differing properties. In the paragraph prededing Theorem 3, similar comments were made for ∇(Γ′), where Γ′ is the dodecahedral graph or the Coxeter graph. Recall that the union of two 1-factors in the first case is hamiltonian while in the second case it is not. In Fig. 2(d), the three color partitions of Q4, namely (12)(34), (13)(24) and (14)(23), 1 yield corresponding 2-factorizations with two 2-factors of equal length 2 |V (Q4)| = 8 each. We denote this facts by writing Q4(2, 2, 2). In similar fashion, we can denote items in Fig. 2 as 4 4 3 0 follows: (e) {4, 4}12,2(1, 4, 3); (f) {4, 4}10,2(1, 2, 1); (g) {4, 4}6,3(3, 3, 3); (h) {4, 4}20,1(2, 1, 1); 0 5 (i) {4, 4}28,1(1, 1, 2); (j) {4, 4}22,1(1, 2, 1). Table IX lists various cases of Theorem 4 item 3(d), indicating without parentheses or commas the triples abc corresponding to the numbers a, b and c of cycles (of equal length in each case) of the respective 2-factors (12), (13) and (14). While the first and third of these 2-factors, namely (12) and (14), can be completed to 2-factorizations composed by 1-zigzagging cycles of equal length (i.e., composed by alternating horizontal and vertical edges), the second 2-factor, (13) is composed by vertical edges and its complementary 2- factor, namely (24), is composed by horizontal edges; while vertical edges form gcd(r, s) cycles of equal length, horizontal edges form t cycles of a different length, so the notation in the previous paragraph cannot be carried out for item 3(d) because gcd(r, s) =6 t. So we s modify that notation by simply writing {4, 4}r,t(a, b, c), that we call the star notation [13]. TABLE IX s =1 5 7 9 11 13 15 s =3 5 7 9 s =5 5 7 9 r =6 113 r =6 131 r =8 211 114 r =8 114 211 r = 10 111 111 511 r = 10 111 511 111 r = 10 555 151 151 r = 12 213 312 112 116 r = 12 132 233 336 r = 12 611 611 211 r = 14 111 111 111 111 117 r = 14 141 111 111 r = 14 711 111 117 r = 16 211 114 114 112 211 118 r = 16 411 112 112 r = 16 811 211 211 r = 18 113 311 111 311 311 111 r = 18 131 131 333 r = 18 911 111 111 r = 20 211 112 215 215 112 112 r = 20 211 512 112 r = 20 a51 251 251 r = 22 111 111 111 111 111 111 r = 22 111 111 111 r = 22 b11 111 111 r = 24 213 314 411 611 611 114 r = 24 133 231 336 r = 24 c11 211 211 r = 26 111 111 111 111 111 111 r = 26 111 111 111 r = 26 d11 111 111 r = 28 211 112 211 211 217 711 r = 28 112 211 112 r = 28 e11 211 217 r = 30 113 311 115 115 111 111 r = 30 132 135 333 r = 30 f11 151 111

s Theorem 21. In the star notation, each case {4, 4}r,t of Theorem 4 that employs just two colors on horizontal paths and the remaining two colors on vertical paths, is expressible as: s 1 1 {4, 4}r,t( 2 gcd(r, t − s),gcd(r, s), 2 gcd(r, t + s)). 1 1 Corollary 22. If 2 gcd(r, t − s) = gcd(r, s) = 2 gcd(r, t + s)=1, then the 2-factors (12), (34), (13), (14) and (23) are composed by a Hamilton cycle each (a total of ﬁve Hamilton cycles), comprehending the 2-factorizations (12)(34) and (14)(23).

27 Additional examples of these statements are provided in Table XI for ﬁxed t = 2, as in Theorem 4 item (3)b. The reader is invited to do similarly for Theorem 4, items 3(a) and 3(c). TABLE X r s =4 s =6 r s =4 s =6 r s =4 s =6 8 141 14 121 121 20 141 122 10 121 16 141 124 22 121 121 12 143 162 18 123 131 24 143 164

The toroidal graphs in Theorem 4 items 1 and 2(b), behave diﬀerently from those in Theorem 21 in that the 2-factors in question are 1-zigzagging as above in only one of the three 2-factorizations, while the other two 1-factorizations are 2-zigzagging (i.e., composed by alternating horizontal and vertical 2-paths), namely: (a) in Theorem 4 items 1-2 just for s ≡ 1 (mod 4), the 1-factorization (12)(34) is 1-zigzagging and the 1-factorizations (13)(24)–(14)(23) are 2-zigzagging; (b) in Theorem 4 item 2(b) just for s ≡ 3 (mod 4), the 2-factorizations (12)(34)–(13)(24) are 2-zigzagging and the 2-factorization (14)(23) is 1-zigzagging. Corollary 23. The graphs Γ in Theorem 4: items 1 and 2(a), are Γ(2, 2, 2); item 2(b) just for s ≡ 1 (mod 4) are Γ(211); and item 2(b) just for s ≡ 3 (mod 4) are Γ(112). Moreover, in all cases of Theorem 4 item 2(b), there are exactly two Hamiltonian 2-factorizations.

9 Applications to 3-dimensional geometry

Figure 14: Two egc edge-colored 4-cube graphs

Fig. 14 redraws the two toroidal copies of Q4 in the lower center–right of Fig. 3, (which arise respectively in Subsection 3.1 from the central–right latin squares in display (1)), as

28 edge-colored tesseracts, in order to extract piecewise linear (PL)[25] realizations of two enantiomorphic compounds of four M¨obius strips each [15, 16]. In the sequel, this results to be equivalent to corresponding enantiomorphic Holden-Odom-Coxeter polylinks of four locked hollow equilateral triangles each [9, 17, 18], from a group-theoretical point of view .

9.1 Usage of the two nontrivial Latin squares in MOLS(4)

Fig. 15 contains two horizontal sets of four copies of Q3 each. A 6-cycle ξi is distinguished i in each such copy Q3 of Q3 with a respective color i (=1 for red, =2 for blue, =3 for green i j and =4 for hazel). Each Q3 has its ξi with the edges marked xi , where x = t for a trapezoid and x = p for a parallelogram (quadrangles that are the faces of PL M¨obius strips as in Fig. 18), i ∈{1, 2, 3, 4} for associated color and j ∈{h,v,d} for edge direction (h for horizontal, i v for vertical and d for in-depth). The top (resp. bottom) set has each Q3 with a 1-to-1 j j correspondence from its edges ti (pi ) to the edges with color i and direction j in the inner i (outer) copy of Q3 in the left (resp. right) copy of Q4 in Fig. 14. This way, each Q3 \ ξi in Fig. 15 is formed by two antipodal vertices that are joined by a color-i dashed axis, allowing to visualize 120◦ angle rotations representing color-and-(t-or-p)-preserving automorphisms that will reappear in relation to Fig. 17-18.

Figure 15: Edges of rotation

The two leftmost 3-cubes in Fig. 16 have each of their edges assigned a pair of colors in {1, 2, 3, 4}. The ﬁrst (resp. second) color is obtained from the corresponding row of 3-cubes j j in Fig. 15 as the color i of the only edge ti (resp. pi ) in that edge position among the four cases in the row. This allows an assignment of a color to each face quadrant of the cube, as shown on the center and right in Fig. 16, and ﬁguring the 24 such quadrants in each of the two cases. Note that the faces in these cubes are given an orientation each. The color pairs in Fig. 15 can be recovered from the quadrant colors in Fig. 16 by reading them along an edge according to the corresponding face orientation.

29 9.2 Compound of four PL Möbius strips Fig. 17 depicts the union [0, 3]3 ⊂ R3 of 27 unit 3-cubes. In it, information carried by the top rows in Fig. 15-15 determines four PL trefoil knots, one per each color i ∈{1, 2, 3, 4}. These trefoil knots are indicated in thick trace along edges of the said unit cubes. These i-colored thick-traced edges determine two parallel sides of either a trapezoid or a parallelogram, as illustrated in Fig. 18, for i = 4 = hazel color. In the case of a trapezoid (resp. parallelogram), the lengths of those sides are 3 internally and 1 externally (resp. 2 and 2). The other two sides of each trapezoid or parallelogram are presented in thin trace in the color i to distinguish them from the thick trace of the trefoil knot sides. Note that the constructed trapezoids and parallelograms determine four PL Möbius strips that give place to the following results. (Similar results for the bottom rows in Fig. 15-15 are omitted and left for the interested reader to figure out).

Figure 16: Bilabels

3 Theorem 24. There exists a maximum-area PL M¨obius strip M1 embedded in [0, 3] \ 3 3 [1, 2] ⊂ R whose boundary is a PL closed curve C1 formed by a minimum of segments parallel to the coordinate directions and whose end-vertices are points in Z3.

Proof. A strip as in the statement is represented in Fig. 18 (whose PL boundary is also present in Fig. 17 as the hazel trefoil knot), with the mentioned segments as in the following display (where, starting clockwise at the left upper corner of Fig. 18, colors of quadrilaterals

30 are cited and, between parentheses, whether they are shown full or in part):

gray(part) green(part) yellow(full) gray(part) green(full) yellow(part) ======h v d h v d (3) [103, 203]1 [203, 201]2 [201, 231]3 [231, 031]2 [031, 032]1 [032, 012]2 h v d h v d [012, 312]3 [312, 310]2 [310, 320]1 [320, 120]2 [120, 123]3 [123, 103]2

Figure 17: PL M¨obius strip whose PL boundary has color 4 = hazel

Here, a segment denoted [a1a2a3, b1b2b3] stands for [(a1, a2, a3), (b1, b2, b3)], with ai, bi ∈ {0, 1, 2, 3}, (i = 1, 2, 3). In (1), each of the 12 segments are appended with its length as a subindex and an element of {h,v,d} as a superindex, where h, v, and d stand for horizontal, vertical and in-depth directions, respectively. The PL curve C4, a PL trefoil knot, is depicted in thick hazel trace in Fig. 17-17 (in Fig. 17 via unbroken unit-segment edges).

Theorem 25. The maximum number of M¨obius strips in [0, 3]3 \ [1, 2]3 as in Theorem 24 and whose boundaries have pairwise intersections of dimension 0 (i.e., isolated points) is 4.

Proof. To prove the statement, consider Fig. 17, where C1 is again drawn, as well as the blue PL trefoil C2, deﬁned by the segments:

31 h 2 d h v d [100, 200]1 [200, 220]v [220, 223]3 [223, 023]2 [023, 013]1 [013, 011]2 h 2 d h v d (4) [011, 311]3 [311, 331]v [331, 332]1 [332, 331]2 [331, 102]3 [102, 100]2

Figure 18: Four edge-disjoint PL trefoil subgraphs bounding four PL M¨obius strips

and the green PL trefoil C3, deﬁned by the segments:

h v d h v d [133, 233]1 [233, 213]2 [213, 210]3 [210, 010]2 [010, 020]1 [020, 022]2 h v d h v d (5) [022, 322]3 [322, 302]2 [302, 301]1 [301, 101]2 [101, 131]3 [131, 133]2

and the brown PL trefoil C4, deﬁned by the segments:

h v d h v d [230, 130]1 [130, 110]2 [110, 113]3 [113, 313]2 [313, 323]1 [323, 321]2 h v d h v d (6) [321, 021]3 [021, 001]2 [001, 002]1 [002, 202]2 [202, 232]3 [232, 230]2

Observe that the six (maximum) planar faces of M1 intersect C4 in six pairs of segments, h h shown in display (3) vertically: ﬁrst [103, 203]1 above and [012, 312]3 below, which form an v isosceles trapezoid together with the segments [103, 012] and [203, 312]; then [203, 201]2 above v and [312, 310]2 below, which form a parallelogram together with the segments [203, 312] and [201, 310], etc. Similar observations can be made with respect to displays (4-6). We denote h d v h d v such planar faces in (3-6) by ti ,pi , ti ,pi , ti ,pi , (i ∈ {1, 2, 3, 4}), as on the rightmost (color 4 =) hazel 3-cube in the schematic Fig. 15.

32 Let us denote the maximum PL M¨obius strips expanded by the PL trefoil knots C2, C3 and C4 respectively as M2, M3 and M4. Let us look at the segmental intersections of trapezoids and parallelograms in M = M1 ∪M2 ∪M3 ∪M4 from displays (1-4) above . Recall: color 1 is red, color 2 is blue, color 3 is green and color 4 is brown.

Figure 19: Odom-Coxeter polylink

h 3 The top-front set T Fi = ([000, 300] × [000, 010] × [000, 001]) ∩ Mi in [0, 3] is trapezoid h h t2 for i = 2 and parallelogram p3 for i = 3. In the upper leftmost cube in Fig. 16, let us call it Q, we indicate this segmental intersection by the edge-labelling pair 23, with 2 in blue and 3 in green, which are the colors used to represent C2 and C3, respectively. This pair h h 23 labels the top-front horizontal edge in Q, corresponding to the position of T F2 ∩ T F3 in Fig. 17-17. In all edge-labelling pairs in Q (Fig. 16), the first number is associated to a trapezoid and the second one to a parallelogram, each number printed in its associated color. We subdivide the six faces of Q into four quarters each and label them 1 to 4, setting external counterclockwise (or internal clockwise) orientations to the faces, so that the two quarter numbers corresponding to an edge of any given face yield, via the defined orientation, the labelling pair as defined in the previous paragraph. This is represented in the center and rightmost cubes in Fig. 16. We can identify the cube Q with the union of the pairwise distinct intersections Mi ∩ Mj,

33 (1 ≤ i < j ≤ 4). This way, Q becomes formed by the segments:

3 4 2 1 [(0.5,0.5,0.5),(2.5,0.5,0.5)]2,[(0.5,2.5,0.5),(2.5,2.5,0.5)]1,[(0.5,0.5,2.5),(2.5,0.5,2.5)]3,[(0.5,2.5,2.5),(2.5,2.5,2.5)]4, 4 2 3 1 [(0.5,0.5,0.5),(0.5,2.5,0.5)]3,[(2.5,0.5,0.5),(2.5,2.5,0.5)]1,[(0.5,0.5,2.5),(0.5,2.5,2.5)]4,[(2.5,0.5,2.5),(2.5,2.5,2.5)]2, (7) 2 3 4 1 [(0.5,0.5,0.5),(0.5,0.5,2.5)]4,[(2.5,0.5,0.5),(2.5,0.5,2.5)]1,[(0.5,2.5,0.5),(0.5,2.5,2.5)]2,[(2.5,2.5,0.5),(2.5,2.5,2.5)]3, where each segment is suffixed with its trapezoid color number as a subindex and its parallelogram color number as a superindex. The first, second and third lines in (5) display those segments of Q parallel to the first, second and third coordinates, respectively. Fig. 15 represents schematically the Möbius strips Mi in Q, for i = 1, 2, 3, 4, with the i-colored edges corresponding to the trapezoids and parallelograms of each Mi labeled with j symbols xi , where x = t for trapezoid and x = p for parallelogram, and where j ∈{h,v,d} for horizontal, vertical and in-depth edges of Q, respectively.

4 4 Theorem 26. The automorphism group G = Aut(∪i=1Mi) of ∪i=1Mi is isomorphic to Z2 × Z2 × Z2. Moreover, there are only two four-PL-M¨obius-strip compounds, one of which 4 (∪i=4Mi) is the one with PL M¨obius strip boundaries depicted as the PL trefoils in Fig. 17. Furthermore, these two compounds are enantiomorphic.

3 3 Proof. By way of the reﬂection FO of [0, 3] \[1, 2] about the central vertex O = (1.5, 1.5, 1.5) 3 of [0, 3] , each Mi is transformed bijectively and homeomorphically into M5−i, for i = ◦ 2 2 2 3 1, 2, 3, 4. The 180 angle rotations Rd, Rv, Rh of [0, 3] about the in-depth axis x1 = x2 =1.5, the vertical axis x1 = x3 =1.5 and the horizontal axis x2 = x3 =1.5, respectively, form the following respective transpositions on the vertices of [0, 3]3:

((x1, x2, x3), (3 − x1, 3 − x2, x3 )), ∀xi ∈{0, 3}, (i =1, 2, 3) ((x1, x2, x3), (3 − x1, x2, 3 − x3 )), ∀xi ∈{0, 3}, (i =1, 2, 3) and (8) ((x1, x2, x3), (x1, 3 − x2, 3 − x3 )), ∀xi ∈{0, 3}, (i =1, 2, 3) and determine the following permutations of M¨obius strips Mi, (i =1, 2, 3, 4):

(M1, M3)(M2, M4), (M1, M2)(M3, M4) and (M1, M4)(M2, M3). (9)

On the other hand, neither the reﬂections on the coordinate planes at O = (1.5, 1.5, 1.5) nor ◦ 3 3 3 4 the ±90 angle rotations Rd, Rd, Rv, Rv, Rh, Rh preserve the union ∪i=1Mi but yield a four- 4 PL-M¨obius-strip compound that is enantiomorphic to ∪i=1Mi. Moreover, G is generated by FO, Rd, Rv and Rh.

9.3 Polylink of four hollow triangles A sculpture by George P. Odom Jr., represented for our purposes in Fig. 19, had its structure analyzed by H. S. M. Coxeter [9], who established its relevant geometric and symmetric properties. According to a footnote in [26] pg. 270, Odom and Coxeter were unaware of the earlier discovery [17] of this structure by Alan Holden, who called it a regular polylink of four locked hollow triangles [18].

34 We relate the top M¨obius-strip compound above to this structure, noting that the centers of the maximum linear parts of the PL trefoil knots Ci (i =1, 2, 3, 4) are the vertices of four corresponding equilateral triangles, namely (in the order of the triangle colors): 1 (1.5, 0, 3), (3, 1.5, 0), (0, 3, 1.5); 2: (1.5, 0, 0), (0, 1.5, 3), (3, 3, 1.5); (10) 3: (1.5, 3, 3), (0, 1.5, 0), (3, 0, 1.5); 4: (1.5, 3, 0), (3, 1.5, 3), (0, 0, 1.5).

Each of these equilateral triangles, denoted Ti (i = 1, 2, 3, 4), gives place to a corresponding hollow triangle (i.e., a planar region bounded by two homothetic and concentric ′ equilateral triangles [9]) by removing from the interior of Ti the equilateral triangle Ti whose vertices are the midpoints of the segments between the vertices of Ti (as presented in display (10)) and O = (1.5, 1.5, 1.5). Characterized by their colors, these midpoints are, respectively:

1: (1.50, 0.75, 2.25), (2, 25, 1.50, 0.75), (0.75, 2.25, 1.50); 2: (1.50, 0.75, 0.75), (0.75, 1.50, 2.25), (2.25, 2.25, 1.50); (11) 3: (1.50, 2.25, 2.25), (0.75, 1.50, 0.75), (2.25, 0.75, 1.50); 4: (1.50, 2.25, 0.75), (2.25, 1.50, 2.25), (0.75, 0.75, 1.50). Notice that the centers in display (10) are the vertices of an Archimedean cuboctahedron. We also consider the midpoints of the sides of the four triangles Ti (i =1, 2, 3, 4), namel:

1: (2.25, 0.75, 1.50), (1.50, 2.25, 0.75), (0.75, 1.50, 2.25); 2: (0.75, 0.75, 1.50), (1.50, 2.25, 2.25), (2.25, 1.50, 0.75); (12) 3: (0.75, 2.25, 1.50), (1.50, 0.75, 0.75), (2.25, 1.50, 2.25); 4: (2.25, 2.25, 1.50), (1.50, 0.75, 2.25); (0.75, 1.50, 0.75).

By expressing the 3-tuples in (11) via a 4 × 3-matrix {Ai,j; i = 1, 2, 3; j = 1, 2, 3, 4}, we have the following correspondence from (11) to (12), expressed in terms of the notation of points in Fig. 19:

A1,1 A1,2 A1,3 B1,12 B1,23 B1,31 A3,3 A4,1 A2,2 A A A  B B B  A A A  2,1 2,2 2,3 → 2,12 2,23 2,31 = 4,3 3,1 1,2 (13) A3,1 A3,2 A3,3 B3,12 B3,23 B3,31 A1,3 A2,1 A4,2       A4,1 A4,2 A4,3 B4,12 B4,23 B4,31 A2,3 A1,1 A3,2 with the following meaning: each midpoint Bi,jk of a large triangle Ti between vertices Bi,j ′ ′ ′ and Bi,k coincides with a corresponding vertex Ai ,j of some small triangle Ti′ , and vice-versa, as depicted in Fig. 19. Now, we take each Ti (i =1, 2, 3, 4) as the 6-cycle of its vertices and side midpoints:

T1 = (B1,1 A3,3 B1,2 A4,1 B1,3 A2,2); T = (B A B A B A ); 2 2,1 1,2 2,3 3,1 2,2 4,1 (14) T3 = (B3,1 A1,3 B3,2 A2,1 B3,3 A4,2); T4 = (B4,1 A1,1 B4,2 A2,3 B4,1 A3,2). 4 4 Corollary 27. The automorphism group of the union ∪i=1Ti is G = Aut(∪i=1Mi) = 4 Aut(∪i=1Ti). In addition, Gi = Aut(Mi) = Aut(Ti), for each i = 1, 2, 3, 4, is isomorphic to the dihedral group of six elements. Moreover, there are only two polylinks of four 3 4 hollow triangles in [0, 3] , including ∪i=1Ti, and these two polylinks are enantiomorphic.

35 Proof. By expressing, as in Fig. 19, the vertices of [0, 3]3 by:

0 = 000, 1 = 300, 2 = 030, 3 = 330, 4 = 003, 5 = 303, 6 = 033, 7 = 033, (15) we notice that the ±120◦ angle rotations of [0, 3]3 around the axis line determined by the two points of color i in Q, as indicated in Fig. 15, correspond respectively to the permutations:

−1 Color 1 : R1 =(124)(365) and R1 =(q142)(563); −1 Color 2 : R2 =(036)(174) and R2 =(063)(147); −1 (16) Color 3 : R3 =(065)(271) and R3 =(056)(217); −1 Color 4 : R4 =(247)(053) and R4 =(274)(035), where color 1 is red, color 2 is blue, color 3 is green and color 4 is hazel; notice also that the axes in Q corresponding to these colors are: [(0.5, 0.5, 0.5), (2.5, 2.5, 2.5)], [(0.5, 2.5, 0.5), (2.5, 0.5, 2.5)], [(0.5, 0.5, 2.5), (2.5, 2.5, 0.5)], [(2.5, 0.5, 0.5), (0.5, 2.5, 2.5)]. The rest of the statement arises from Theorem 26. The fact that there are just two polylinks of four hollow triangles in [0, 3]3 and that they are enantiomorphic is inherited from Theo- rem 26. Acknowledgement: The author is thankful to an anonymous reviewer of a previous draft for an interesting and educative report and for the counterexamples and corrections indicated in it, allowing to rectify the exposition of ideas in the present version.

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