3-maps David L. Craft* Muskingum College Arthur T. White Western Michigan University

Consider the standard die. The face numbers are arranged around each vertex —from smallest to largest— in either clockwise (solid dot) or counterclockwise (hollow dot) fashion. There are four of each type of vertex Any numbering of the faces, from 1 to 6, results in four of each type of vertex. Any numbering of the faces of a tetrahedron, from 1 to 4, or a dodecahedron, from 1 to 12, exhibit this balance. Vertex type is undefined for the octahedron and the icosahedron where the degree of regularity is not 3.

Theorem 1 For a cubic map on S n with regions properly colored with integers, the number of clockwise vertices equals the number of counterclockwise vertices. We consider maps of cubic graphs 2-cell imbedded on closed orientable 2-manifolds S n, where n is a nonnegative integer

A 3-map is a 3-chromatic map M of a cubic graph G on a surface S n, including n = 0 for plane cubic graphs. Ex. Q 3 as a 3-mapor Interesting Properties: The map is 3-region colored (i.e., it is a 3-map) The graph is bipartite (given by the order of colors around each vertex) The map is 1-factorable; i.e., 3-edge colorable where each edge receives the unique color not assigned to either region it bounds The vertex set has three partitions, one for each region color class.

A Few Historic Results with Variations. Theorem A (Konig): A cubic bipartite graph is 3-edge colorable. Theorem B (Grotzsch): Every plane graph with no triangles is 3-vertex colorable. Theorem C: If every cubic plane map is 4-region colorable, then every plane map is 4-region colorable. Theorem D (Tait): Let G be a bridgeless cubic plane graph. Then G can be 3-edge colored if and only if G can be 4-region colored. Theorem E (Tait): Let G be a cubic graph (not necessarily planar). Then G can be 3-edge colored if and only if G is spanned by a collection of disjoint cycles of even length. Theorem F: Let G be a bridgeless cubic plane graph. Then G can be 4-region colored if and only if G is spanned by a collection of disjoint cycles of even length. (This follows from D and E.)

More history … Theorem G’ (Heawood--dual form): A plane cubic map can be 3-region colored if and only if all regions have even length. Lemma: A plane graph is bipartite if and only if all region lengths are even. Theorem G’’: A plane cubic map can be 3-region colored if and only if it is bipartite. Conjecture H’ (Grunbaum—dual form): A cubic graph on S n can be 3-edge colored, provided that the dual has neither loops nor multiple edges. Conjecture I (Tutte): If G is cubic, with no loops or multiple edges, and no bridges and no subdivision of the Petersen graph as a subgraph then G can be 3-edge colored.

Characterizing 3-maps Theorem 2: Let M be a map for a cubic graph G. Then the following are equivalent: (a)M is 3-region colorable (i.e., M is a 3-map). (b)G is bipartite, and the bipartition is canonical (i.e., the partite sets are given by one of two possible color rotations at each vertex). (c)G is canonically 3-edge colorable (with opposite color rotations at the endpoints of each edge), and every bi-colored cycle bounds a region. Moreover, every region is so described. (d)S(G) = {(u, v): uv is in E(G)} is partitioned into three sets, each including a collection of region- bounding directed cycles partitioning V(G).

Theorem 3: A connected cubic graph is the underlying graph for some 3-map if and only if it is bipartite. Idea of Proof: If G underlies a 3-map, it is bipartite by (b) of Theorem 2. Conversely, if G is connected, cubic, and bipartite it has a 3-edge coloring Theorem A. Assume such a coloring and use opposite color rotations on the partite sets to define the imbedding.

Example: is cubic, bipartite, and has several 3-edge colorings The “usual” coloring gives a planar 3-map. Switching the colors on one 4-cycle (top) gives a 3-map on S 1 Switching the colors on one 6-cycle instead gives a 3-map on S 2 Use your imagination. (You can trace four regions: two 12-cycles and two 6-cycles.) CONSTRUCTING 3-MAPS A. The prisms

CONSTRUCTING 3-MAPS B.Stellate/Truncate Begin with a bipartite map.

CONSTRUCTING 3-MAPS B.Stellate/Truncate Stellate each region: Place a vertex in the interior and join it to all boundary vertices.

CONSTRUCTING 3-MAPS B.Stellate/Truncate Stellate each region: Place a vertex in the interior and join it to all boundary vertices.

CONSTRUCTING 3-MAPS B.Stellate/Truncate Truncate each vertex: Place a cycle around it

CONSTRUCTING 3-MAPS B.Stellate/Truncate Truncate each vertex: Place a cycle around it and delete its interior.

CONSTRUCTING 3-MAPS B.Stellate/Truncate Result: A bipartite map

CONSTRUCTING 3-MAPS B.Stellate/Truncate Result: A bipartite map that is 3-region colorable. A 3-map!

CONSTRUCTING 3-MAPS C1. Extend an existing 3-map internally: Doubly subdivide a pair of like-colored edges bounding the same region. Then join these four new vertices as a 4-cycle and extend the original coloring.

CONSTRUCTING 3-MAPS C2. Extend a pair of 3-maps Externally: 3-map on S n 3-map on S m Begin with two 3-maps on their respective surfaces. Select an edge from each and recolor if necessary so the selected edges are the same color.

CONSTRUCTING 3-MAPS C2. Extend a pair of 3-maps Externally: 3-map on S n 3-map on S m Join the two surfaces with a tube attached near the two selected edges in like-colored regions. Then add two edges to join the four new vertices into a 4-cycle and color as before.

CONSTRUCTING 3-MAPS – WITH A VIEW TOWARD GENUS D.Observe that K 3,3, imbedded with 3 hexagons on S 1 is the smallest order 3-map (p = 6). n copies of K 3,3 on S 1, joined together using C2, yields a 3-map on S n

M CONSTRUCTING 3-MAPS – WITH A VIEW TOWARD GENUS E.Expanding on this idea, we can construct a concatenation G*H of a 3-map M for cubic graph G and a connected, bipartite graph H. H MM G*H as a 3-map If H has p vertices and q edges and M is on S m then G*H is on S pm+q-p+1 Note (i) the graph G*H is not well defined and (ii) mirror images of M are used according to the partite set in H. M

Realizability: We call an ordered pair (p,n) realizable if there exists a 3-map of order p on S n ? Question: Which ordered pairs (p, n) are realizable ? Theorem 4: Let p be even. The ordered pair (p, 0) is realizable if and only if p = 8 or p ≥ 12. Idea of the proof: Begin with the 3-map given and expand internally begin with Q3 on S0, denoted M 1 For begin with denoted M 2 For

Theorem 5: Let p be even. For n ≥ 1, (p, n) is realizable if and only if p ≥ 4n+2. Idea of Proof: This is very similar to Theorem 4, using the following two 3-maps as bases and again expanding internally. 1 2n+1 This voltage graph, over the group  = Z 4n+2 lifts to a 3-map of order 4n+2, 2-cell imbedded on S n. Denote this map M 4n This voltage graph, over the group  = Z 4n+4 lifts to a 3-map of order 4n+4, 2-cell imbedded on S n Denote this map M 4n+4

Region distributions A multiset of p/2 + 2 – 2n even integers, each at least 4, is said to be feasible for (p, n) if it can be partitioned into three sub-multisets, each summing to p. Ex. {6 x 4, 3 x 6} is both feasible and realizable for (14, 0) by M 2. Its color classes have regions with sizes {4, 4, 6 ; 4, 4, 6 ; 4, 4, 6} Ex. {7 x 4, 1 x 8} is feasible for (12, 0) as {4, 4, 4 ; 4, 4, 4 ; 4, 8} but it is not realizable as no 3-map of order 12 exists on the sphere whose color-classes have this size distribution. Open problem : Characterize feasible multisets that are realizable.

m-uniform 3-maps A map is said to be m-uniform if all its regions are m-gons. If a 3-map is m-uniform, then each color class consists of k m-gons, where k is a fixed positive integer for which p = mk. We call k the partition size of the map. n = 1 – (6 – m )k / 4 Remark 1: The only 4-uniform 3-map is Q 3 on S 0 (since 2 is the only value of k that gives a non-negative value for n). Observe, an m-uniform 3-map with p = mk vertices has q = 3mk/2 edges and r = 3k regions. If this 3-map is on Sn then mk – 3mk/2 + 3k = 2 – 2n so Remark 2: A 6-uniform 3-map must be on the torus S1. There are infinitely many of these.

General Question: For which values of m and k is there an m-uniform 3-map with partition size k ? Some specific answers: (1)For each positive integer n, M 4n+2 is (4n+2)-uniform with k = 1. (2)The only 4-uniform 3-map is M 1 (i.e., Q 3 ) with k = 2. (3)For each positive integer k, there is a 6-uniform 3-map on S 1 having partition size k. (4) For each positive integer s, there is a (4s)-uniform 3-map of order 8s and hence k = 2, on S 2s-2. (5) For each positive integer s, there is a (4s+2)-uniform 3-map of order 8s + 4 and hence k = 2, on S 2s-1. (6) For each positive integer k, there is a (2k)-uniform 3-map having partition size k, hence order 2k 2 on S (k-1)(k-2)/2

Genus Constructions Theorem 6: Let G be an order p(G) graph for a 3-map M and let H be a connected bipartite, cubic graph with order p(H). Then  (H x G) = 1 + p(H)p(G)/4 Idea of Proof: --Associate a copy of G with each vertex of H—using mirror images of G corresponding to one of the partite sets of H. --By Theorem A, H is 3-edge colorable so assume such a coloring. --For each edge e = uv of H, which is colored i, use a tube to join all corresponding pairs of i-colored regions of the copies of G associated with u and v. --Imbed n edges in each tube joining n-gons, joining corresponding vertex pairs

Genus Constructions Theorems 7 & 8: Let G be a cubic graph of order p having a 3-map M. Then  (G x G) = 1 + p 2 /4. This generalizes to  (G m ) = 1 + p m ((3m - 4)/4) Example: Let G = Q 3 and consider the two different 3-maps for G. By Theorem 7,  (Q3 x Q3) = 17 (1)On S 0, M has six quadrilateral regions—partitioned into 3 color classes of 2 regions each. Each of the 12 edges of Q 3 corresponds to two tubes (joining the two regions of a color class). So there are 24 tubes joining 8 spheres, giving S 17. (2)On S 1, M has two quadrilaterals (one color class) and two octagonal regions (each a color class). Each of the four 1-colored edges of Q 3 corresponds to two tubes. Each of the remaining eight edges of Q 3 corresponds to one tube. So there are 16 tubes joining 8 copies of S 1, giving S 17.

n-maps One natural generalization of 3-maps are imbedded, bipartite n-regular graphs that admit a region coloring with n colors for which all vertices in one partite set have identical color rotations and all vertices in the other paritite set have the opposite color rotation. These are called n-maps. (i) (i+3) (i+2) (i+4) (i+1) (i+2) (i+3) (i+4)

n-maps Example: K 4,4 on S 1 as a 4-map

n-maps A Small Sampling of Theorem Analogues Theorem 10: A connected n-edge colorable, n-regular graph is the Underlying graph for some n-map if and only if it is bipartite. Theorem 11: If M is an n-map on S k with underlying graph G having r i regions of color i, then (a) There is an (n+1)-map on S 2k+ri-1 with underlying graph G x K 2 (b) There is a 4-uniform 2n-map on S 1+(p^2)(n-2)/4 with underlying graph G x G.

n-maps Say the ordered pair (n, m) is realizable if there exists an n-map on S k Theorem 12: If (n, m 1 ) and (n, m 2 ) are realizable then (n, m 1 + m 2 ) is realizable. Theorem 13: (4, m) is realizable if and only if m ≥ 1. Theorem 14: (5, m) is realizable if and only if m ≥ 3. Theorem 15: (6, m) is realizable if and only if m ≥ 4. Open Problem: Generalize these results to (n, m) is realizable if and only if m ≥ f(n).