1. Multicolored Subgraphs in Properly Edge-colored Graphs
Hung-Lin Fu (傅 恒 霖)
Department of Applied Mathematics
National Chiao Tung University
Hsin Chu, Taiwan 30010

2. Goals
The subject we study is to find certain special subgraphs inside a properly edge-colored graph; the coloring may be prescribed or arbitrarily given.
A good place to start is considering the complete graphs which are properly edge-colored with minimum number of colors, their chromatic indices.

3. Preliminaries
A (proper) k-edge-coloring of a graph G is a mapping from E(G) into {1, . . ., k } such that incident edges of G receive (distinct) colors.
In this talk, all colorings are proper, i. e. incident edges receive distinct colors.
A proper 3-edge-coloring of K4

4. Chromatic Indices
It is well-known that the complete graph of order n, Kn, is of Class 1 if and only if n is even.
That is, ’(Kn) = n – 1 if n is even and ’(Kn) = n if n is odd.
A graph G is of Class 1 if its chromatic index is equal to ∆(G) its maximum degree.

5. Multicolored graph
A subgraph H in an edge-colored graph G is a multicolored subgraph of G if no two edges in H have the same color. (A multicolored graph is also known as a rainbow, hetero-chromatic or poly-chromatic graph.) 1 2 3 4 5 1 2 3 4 5
A multicolored 5-cycle in a properly edge-colored K5

6. Motivation
Theorem (Woolbright and Fu, 1998) In any (2m-1)-edge-colored K2m where m > 2, there exists a rainbow 1-factor.
This result has been further improved to complete uniform hypergraphs by Saad El-Zanati et al. later.
It remains unknown for finding two or more edge-disjoint rainbow 1-factors.

7. Follow-ups
David looked at finding a long rainbow path, hopefully a Hamilton rainbow path.
He believed that a fairly long rainbow path can be found. (There are results on this part.)
I looked at finding more rainbow 1-factors which are edge-disjoint.
I can not find even two such 1-factors after 20 years of “looking” if the coloring is arbitrarily given.

8. More Rainbow 1-factors
If we are allowed to design an edge-coloring for K2m, then we are able to find (2m – 1) rainbow 1-factors inside K2m.
This can be done by simply constructing a room square of order 2m, see example of order 8.

9. Room square of order 8 7 edge-disjoint rainbow matchings! 1,8 5,7 3,4
2,6
3,7
2,8
6,1
4,5
5,6
4,1
3,8
7,2
6,7
5,2
4,8
1,3
2,4
7,1
6,3
5,8
3,5
1,2
7,4
6,8
4,6
2,3
1,5
7,8
7 edge-disjoint rainbow matchings!

10. Multicolored subgraphs
For convenience in locating the references in literatures, we shall use Multicolored subgraphs in what follows.
Of course, Rainbow is more beautiful!

11. Taipei City with two rainbows

12. Prescribed coloring
Constantine’s Weak Conjecture (2002) For any m > 2, K2m can be (2m－1)-edge-colored in such a way that the edges can be partitioned into m isomorphic multicolored spanning trees.
This conjecture has been verified later in (Work jointly with S. Akbari, A. Alipour and Y. H. Lo.)

13. Isomorphic multicolored trees in K6
T T T3
Color 1 x3x5 x4x6 x1x2
Color 2 x2x4 x1x5 x3x6
Color 3 x2x5 x3x4 x1x6
Color 4 x2x6 x1x3 x4x5
Color 5 x1x4 x2x3 x5x6 x1 x2 x3 x4 x5 x6 T1 3 2 4 5 1

14. Preliminaries
A k-total-coloring of a graph G is a mapping from V(G)∪E(G) into {1,…,k } such that : (1) adjacent vertices in G receive distinct colors, (2) incident edges of G receive distinct colors, and (3) any vertex and its incident edges receive distinct colors. 1 2 3 4 5 1 4 2 5 3 ≡
5-total-colored K5

15. An example: m = 5
Bipartite difference : 1, 2, 3 and 4

16. An example: m = 5 8 6 4 2
Tree 1 5 1 3 7 9

17. An example: m = 5
Tree 2

18. On K2m+1 Case
Constantine’s Weak Conjecture on odd order (2005) For any m ≥ 2, K2m+1 can be (2m+1)-edge colored in such a way that the edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs.
Later, we (with Y. H. Lo) verify this conjecture by decomposing K2m+1 (with designed coloring) into multicolored Hamilton cycles.
All 2m+1 colors occur in a Hamilton cycle.

19. If the edge-coloring is not designed, then finding multicolored spanning trees is not that easy except the first one!

20. Multicolored Spanning Trees
Brualdi-Hollingsworth’s Conjecture (1998, JCT(B)) If m > 2, then in any proper (2m－1)-edge-coloring of K2m, all edges can be partitioned into m multicolored spanning trees. (They found two.)
Theorem (Krussel, Marshall and Verall, 2000, Ars Combin.) Three edge-disjoint multicolored spanning trees always exist.

21. Conjectures
Constantine’s Strong Conjecture (2002) If m > 2, then in any properly (2m－1)-edge-colored K2m, all edges can be partitioned into m isomorphic multicolored spanning trees.
Constantine’s Strong Conjecture on odd order (2005) In any properly (2m+1)-edge-colored K2m+1, all edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs.
Unicyclic graph: a graph with exactly one cycle.

22. Observation
The above two conjectures are too hard to be verified in my opinion.
There is a chance to get something done on the conjecture posed by Brualdi and Hollingsworth.
Find more (＞3) multicolored spanning trees!

23. Recent progress
By using probabilistic techniques, Carraher et al. prove that about m/(500log(2m)) mutually edge-disjoint rainbow spanning trees exist for m ≥ 500,000.
We construct at least [(6m+9)1/2/3] such trees by using an explicit construction.
This is a joint work with Yuan-Hsun Lo, K. E. Perry and C. A. Rodger.

24. Algorithmic approach
The first tree we can get is a spanning star with the center called root.
In order to find the second one, we have to make a choice of the second root. Clearly, we have to make an adjustment of the first tree. Otherwise, the second tree will not be a spanning tree.
So, to search for the (k+1)th tree we adjust the previous all k trees.
We stop when there is no way to adjust previous ones. (Our work depends on the number of pendant leaves.)

25. Edge-switching
φ is a (2m–1)-edge-coloring of K2m, and T is a multicolored spanning tree with root x. If x is incident to two leaves e1 = xy1 and e2 = xy2, then let T[x; y1, y2; z1, z2] = T – e1 – e2 + y1z1 + y2z2, for some z1, z2. (Obtained by switching e1 and e2 with y1z1 and y2z2 respectively.)
Fact:
If φ(e1) = φ(y2z2) and φ(e2) = φ(y1z1),
then T[x; y1, y2; z1, z2] is multicolored.

26. MST x e2 e1
T[x; y1, y2; z1, z2] y1 z2 y2 z1

27. Can we add one more?
In order to find a new one, we first “revise” the multicolored spanning trees which have been obtained earlier and then define a new one accordingly.
This depends on the number of pendant leaves which occurred in previous trees.
The argument for trueness is somehow very complicate.

28. Quick note
If we have k edge-disjoint spanning trees, then the maximum degree of each tree is at most 2m - k.
In our construction, this value is 2m – 2k since we use edge-switching.
Hence, our algorithm can only go this far.
A smarter idea is need to improve this result.

29. For K2m+1 case
By a similar argument with one pendant vertex fixed, we are able to find around the same number of multicolored spanning unicyclic subgraphs.
Clearly, there is room for improvement; O(m1/2) is still far from m.

30. Smaller multicolored isomorphic subgraphs
It is easy to see that if Kn (with designed or un-designed edge coloring) can be decomposed into C3’s, then all C3’s are multicolored.
But, it is not that trivial for C4-decomposition even we can design the edge coloring.
We expect all cycle decompositions of admissible order and a designed edge coloring, a multicolored cycle design can be obtained.(?)
This is what I am working now: Rainbow graph designs.

31. Examples (Designed coloring)
A 4-cycle design of order n exists if and only if n ≡ 1 (mod 8).
For each odd integer n, there exists an n-edge coloring (Class 2) of Kn.
The coloring can be designed as follows.
Let V(Kn) = Zn and the edge {x, y} receives the color “i” if x + y is congruent to 2i modulo n. (We use color n for 0.) 1 1 2 3 4 5 2 5 3 4

32. Cyclic multicolored 4-cycle system
Let n = 8k + 1, k is a positive integer.
The base cycles are: (3, 4i+1, 4, 4i+3), i = 1, 2, …, k.
It is not difficult to check each difference from 1 to 4k occurs exactly once and the colors in the cycle (3, 4i+1, 4, 4i+3) are 2i + 2, 2i + 3, 4k - 2i - 2 and 4k - 2i – 3 and they are distinct.

33. Other cycle-systems?
As we have seen multicolored cycle systems of order n do exist when the cycle lengths are 4 and n.
Problem: Find a multicolored cycle system for each cycle length larger than 4.
Example: For the same edge-coloring, (0, 3, 8, 10, 11, 4) generates a multicolored 6-cycle system of order 13.

34. Follow up (Arbitrarily given coloring)
Can we find a multicolored fairly long path in a properly colored complete graph?
Given an n-edge-colored Kn, a rainbow Hamiltonian path exists for n > n0. (!?) (I believe that this is a true.)
Problem: How many edge-disjoint rainbow matchings of fixed size can we find in an edge-colored complete graph?

35. Multicolored 4-cycles
Given a complete graph of order n = 8k+1 which is n-edge colored. Can we decompose the graph into k(8k+1) multicolored 4-cycles?
Clearly, we have no idea about how the edges are colored, only a proper coloring is known.
We should be able to do it in my opinion, but I am not able to get this job done yet!

36. Multicolored 4-cycle design of Kn,n
If n is even and Kn,n is n-edge-colored, then there exists a multicolored design of Kn,n provided n ≥ 4.
This fact is equivalent to partition a Latin square of order n into n2/4 sub-squares.
An example: 1 2 3 4

37. Multicolored matching in Kn,n
An n-edge-colored Kn,n can be represented as a Latin square of order n.
A multicolored matching is then a partial transversal of the square.
Everyone knows that finding a partial transversal of maximum size in an arbitrary Latin square is a very difficult problem.
Ryser believed the size is n if n is odd and Brualdi conjectured n - 1 in general.
How about smaller matchings?

38. Problems I would like to work!
Prove that an (8k+1)-edge-colored K8k+1 can be decomposed into multicolored 4-cycles, i.e. a multicolored 4-cycle design exists for each admissible order.
Extend the above result to 4-cycle GDD.
How about multicolored k-cycle designs for admissible orders?

39. Reference
[1] S. Akbari, A. Alipour, H. L. Fu and Y. H. Lo, Multicolored parallelisms of isomorphic spanning trees, SIAM J. Discrete Math. 20 (2006), No. 3,
[2] R. A. Brualdi and S. Hollingsworth, Multicolored trees in complete graphs, J. Combin. Theory Ser. B 68 (1996), No. 2,
[3] G. M. Constantine, Multicolored parallelisms of isomorphic spanning trees, Discrete Math. Theor. Comput. Sci. 5 (2002), No. 1,
[4] G. M. Constantine, Edge-disjoint isomorphic multicolored trees and cycles in complete graphs, SIAM J. Discrete Math. 18 (2005), No. 3,

40. Continued
[5] H. L. Fu and Y. H. Lo, Multicolored parallelisms of Hamiltonian cycles, Discrete Math. 309 (2009), No. 14,
[6] H. L. Fu and Y. H. Lo, Multicolored isomorphic spanning trees in complete graphs, Ars Combinatoria, 122 (2015),
[7] H. L. Fu and D. E. Woolbright, On the exists of rainbows in 1-factorizations of K2n, J. Combin. Des. 6 (1998), 1-20.
[8] J. Krussel, S. Marshal and H. Verral, Spanning trees orthogonal to one-factorizations of K2n, Ars Combin. 57 (2000),
[9] H. L. Fu, Yuan-Hsun Lo, K. E. Perry and C. A. Rodger, On the number of rainbow spanning trees in edge-colored complete graphs, in preprints. .

41. Thank you for your attention!