3-maps

David L. Craft*Muskingum College

Arthur T. WhiteWestern 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.

6

1

2

3 4

5

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 Sn 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 Sn , where n is a nonnegative integer.

1

12

3 3

1

2

2

3

A 3-map is a 3-chromatic map M of a cubic graph G on a surface Sn, including n = 0 for plane cubic graphs.

Ex. Q3 as a 3-map or1

1

3

2

2

3

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-vertexcolorable.

Theorem C: If every cubic plane map is 4-region colorable, then everyplane 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). ThenG can be 3-edge colored if and only if G is spanned by acollection 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 ofdisjoint 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 Sn can be 3-edge colored, provided that the dual has neither loops normultiple edges.

Conjecture I (Tutte): If G is cubic, with no loops or multiple edges, and nobridges and no subdivision of the Petersen graph as a subgraphthen 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 coloringTheorem A. Assume such a coloring and use opposite color rotations onthe partite sets to define the imbedding.

62 CK Example:

is cubic, bipartite, and hasseveral 3-edge colorings

The “usual” coloring gives a planar 3-map.

Switching the colors on one 4-cycle (top)gives a 3-map on S1

Switching the colors on one 6-cycle insteadgives a 3-map on S2

Use your imagination. (Youcan trace four regions: two 12-cycles and two 6-cycles.)

CONSTRUCTING 3-MAPS

A. The prisms kCK 22

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 allboundary vertices.

CONSTRUCTING 3-MAPS

B. Stellate/Truncate

Stellate each region: Place a vertex in the interior and join it to allboundary 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 Sn3-map on Sm

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 Sn3-map on Sm

Join the two surfaces with a tube attached near the twoselected 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 K3,3 , imbedded with 3 hexagons on S1 is the smallest order 3-map (p = 6).

n copies of K3,3 on S1 , joined together using C2, yields a 3-map on Sn

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 Mfor cubic graph G and a connected, bipartite graph H.

H

M

M

G*H as a 3-map

If H has p vertices and q edges and M is on Sm then G*H is on Spm+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 Sn ?

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.

1

123 3

1

2

2

3

)4(mod0p begin with Q3 on S0, denoted M1

For )4(mod2p begin with

denoted M2

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-mapsas bases and again expanding internally.

1

2n+1

This voltage graph, over the group = Z4n+2 lifts to a 3-map oforder 4n+2 , 2-cell imbedded on Sn . Denote this map M4n+2

1 3 5This voltage graph, over the group = Z4n+4 lifts to a 3-map oforder 4n+4 , 2-cell imbedded on Sn Denote this map M4n+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 M2. Its color classes have regionswith sizes {4, 4, 6 ; 4, 4, 6 ; 4, 4, 6} 1

123 3

1

2

2

3

Ex. {7 x 4, 1 x 8} is feasible for (12, 0) as {4, 4, 4 ; 4, 4, 4 ; 4, 8} but it is notrealizable as no 3-map of order 12 exists on the sphere whose color-classeshave 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 ism-uniform, then each color class consists of k m-gons, where k is a fixedpositive 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 Q3 on S0 (since 2 is the only value of k that givesa 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, M4n+2 is (4n+2)-uniform with k = 1.

(2) The only 4-uniform 3-map is M1 (i.e., Q3) with k = 2.

(3) For each positive integer k, there is a 6-uniform 3-map on S1 having partition size k.

(4) For each positive integer s, there is a (4s)-uniform 3-map of order 8sand hence k = 2, on S2s-2 .

(5) For each positive integer s, there is a (4s+2)-uniform 3-map of order 8s + 4 and hence k = 2, on S2s-1 .

(6) For each positive integer k, there is a (2k)-uniform 3-map having partition size k, hence order 2k2 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 + p2 /4. This generalizes to

(Gm) = 1 + pm ((3m - 4)/4)

Example: Let G = Q3 and consider the two different 3-maps for G.By Theorem 7, (Q3 x Q3) = 17

(1) On S0 , M has six quadrilateral regions—partitioned into 3 color classes of 2 regions each. Each of the 12 edges of Q3 corresponds to two tubes

(joining the two regions of a color class). So there are 24 tubes joining 8 spheres, giving S17 .

(2) On S1, M has two quadrilaterals (one color class) and two octagonal regions (each a color class). Each of the four 1-colored edges of Q3 corresponds to two tubes. Each of the remaining eight edges of Q3 corresponds to one tube. So there are 16 tubes joining 8 copies of S1 , giving S17 .

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 inone partite set have identical color rotations and all vertices in the otherparitite set have the opposite color rotation. These are called n-maps.

(i)

(i+3)

(i+2)

(i+4)

(i+1)(i+1)

(i+2)

(i+3)

(i+4)

n-maps

Example: K4,4 on S1 as a 4-map

n-maps

A Small Sampling of Theorem Analogues

Theorem 10: A connected n-edge colorable, n-regular graph is theUnderlying graph for some n-map if and only if it is bipartite.

Theorem 11: If M is an n-map on Sk with underlying graph G havingri regions of color i, then (a) There is an (n+1)-map on S2k+ri-1 with underlying graph G x K2 (b) There is a 4-uniform 2n-map on S1+(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 Sk

Theorem 12: If (n, m1) and (n, m2) are realizable then (n, m1 + m2) 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).