SIAM J. DISCRETEMATH. c 2006 Society for Industrial and Applied Mathematics Vol. 20, No. 3, pp. 564–567
MULTICOLORED PARALLELISMS OF ISOMORPHIC SPANNING TREES∗
S. AKBARI†, A. ALIPOUR†, H. L. FU‡, AND Y. H. LO‡
Abstract. A subgraph in an edge-colored graph is multicolored if all its edges receive distinct
colors. In this paper, we prove that a complete graph on 2m (m= 2) vertices K2mcan be properly
edge-colored with 2m− 1 colors in such a way that the edges of K2m can be partitioned into m
multicolored isomorphic spanning trees.
Key words. complete graph, multicolored tree, parallelism AMS subject classifications. 05B15, 05C05, 05C15, 05C70 DOI. 10.1137/S0895480104446015
A spanning subgraph of a graph G is a subgraph H with V (H) = V (G). A proper
k-edge coloring of a graph G is a mapping from E(G) into a set of colors {1, . . . , k}
such that incident edges of G receive distinct colors. An h-total-coloring of a graph G is a mapping from V (G)∪ E(G) into a set of colors {1, . . . , h} such that (i) adjacent vertices in G receive distinct colors, (ii) incident edges in G receive distinct colors, and (iii) any vertex and its incident edges receive distinct colors. The edge chromatic
number of a graph G is the minimum number k for which G has a proper k-edge
coloring. Throughout this paper Km and Km,n denote the complete graph of order
m and the complete bipartite graph with partite sets of sizes m and n, respectively.
It is well known that the edge chromatic number of Kmis m if m is odd, and m− 1
if m is even [7, p. 15]. Assume that m is a natural number. For any integer i we denote the residue of i modulo m in the set{1, . . . , m} by [i]m. The following result
is known.
Lemma 1 (see [7, p. 16]). If m is an odd positive integer, then Kmhas an m-total
coloring.
A Latin square of order m is an m× m array of m symbols in which every symbol occurs exactly once in each row and column of the array. A Room square of side 2m− 1 is a (2m − 1) × (2m − 1) array whose cells are empty or contain an unordered pair of distinct integers chosen from R ={1, . . . , 2m}, such that the entries of a given row contain every member of R precisely once, and similarly for columns, and the array contains every unordered pair of members of R precisely once. Room squares have been found for all odd 2m− 1 ≥ 7 [2, p. 239]. An example of a Room square of side 7 is shown in Table 1.
A subgraph in an edge-colored graph is said to be multicolored if no two edges have the same color. Using a Room square of side 2m− 1 one may obtain a proper ∗Received by the editors September 12, 2004; accepted for publication (in revised form) January
31, 2006; published electronically June 30, 2006.
http://www.siam.org/journals/sidma/20-3/44601.html
†Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran, and Department
of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran (s akbari@sharif.edu, alipour@mehr.sharif.edu). The research of the first and second authors was supported by the Institute for Studies in Theoretical Physics and Mathematics (IPM). The research of the first author was in part supported by a grant from IPM (83050211).
‡Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 30050
(hlfu@math.nctu.edu.tw, yhlo.am93g@nctu.edu.tw). The research of the third and fourth authors was supported by NSC grant 93-2115-M-009-002.
564
MULTICOLORED PARALLELISMS OF ISOMORPHIC SPANNING TREES 565 Table 1 35 17 28 46 26 48 15 37 13 57 68 24 47 16 38 25 58 23 14 67 12 78 56 34 36 45 27 18
edge coloring of K2m with 2m− 1 colors in which all edges can be partitioned into
2m− 1 multicolored perfect matchings. For example, using the rows of Table 1 we give a proper edge coloring of K8 with 7 colors. We denote the vertices of K8 by
1, . . . , 8. In Table 1, if rs appears in the ith row, then we color the edge rs with color i. For instance, the edges 47, 16, 38, 25 are colored with color 4. Each column in Table 1 corresponds to a multicolored perfect matching of K8. In a recent paper
[1] the existence of the multicolored matchings in an arbitrary edge-colored complete graph has been studied. A Latin square of order m corresponds to a proper edge coloring of Km,m with m colors. Indeed if L = (Lij) is a Latin square of order m
and {u1, . . . , um} and {v1, . . . , vm} are two parts of Km,m, then we color the edge
uivj with Lij. Since L has m symbols, we have an m-edge coloring of Km,m, and
since every symbol occurs exactly once in each row and each column of L, the edge coloring is proper. Also the existence of two orthogonal Latin squares of order m corresponds to a proper edge coloring of Km,m with m colors for which all edges can
be partitioned into m multicolored perfect matchings. For example, suppose that
L = (Lij) and R = (Rij) are two orthogonal Latin squares of order m with symbols
of the set {1, . . . , m}, and {u1, . . . , um} and {v1, . . . , vm} are two parts of Km,m. As
we saw before, the function c, where c(uivj) = Lij, is a proper m-edge coloring of
Km,m. For any r, 1≤ r ≤ m, let Mr be the set of all edges uivj such that Rij = r.
Obviously{M1, . . . , Mn} is an edge partition of E(Km,m). Since the symbol r occurs
exactly once in each row and each column of R, Mris a perfect matching, and since L
and R are orthogonal, if Rij = r, then the symbols Lij are distinct and we conclude
that Mr is multicolored. There is a classic result which says that for any natural
number m, m= 2, 6, there exist two orthogonal Latin squares of order m; see [3]. We say that the complete graph K2madmits a multicolored tree parallelism (MTP)
if there exists a proper edge coloring of K2m with 2m− 1 colors for which all edges
can be partitioned into m isomorphic multicolored spanning trees. It is clear that the complete graph K4 does not admit an MTP. We note here that such a partition of
the edges of K2mcan be viewed as a parallelism as defined in [5] by Cameron, with an
additional property due to edge colors. In fact, finding a partition as obtained above corresponds to an arrangement of the edges of K2minto an array of 2m− 1 rows and
m columns such that each row contains the edges with the same color which form
a perfect matching and the edges in each column form a multicolored spanning tree of K2m; moreover, all the m spanning trees are isomorphic. Therefore, the partition
creates a double parallelism of K2m, one from the rows of the perfect matchings
and the other from the columns of the edge disjoint isomorphic spanning trees. The following result has been proven in [6].
Theorem A (see [6]). If m= 1, 3 and K2madmits an MTP, then for any r≥ 1,
K2rmadmits an MTP.
There exist three interesting conjectures on the edge partitioning of the complete graphs into multicolored spanning trees.
566 S. AKBARI, A. ALIPOUR, H. L. FU, AND Y. H. LO Table 2 T1 T2 T3 c1 35 46 12 c2 24 15 36 c3 25 34 16 c4 26 13 45 c5 14 23 56 u u u u u u u u u u u u u u u u u u 5 2 4 6 3 1 1 3 4 2 5 6 5 6 1 3 4 2 T1 T2 T3 Fig. 1.
Constantine’s Conjecture (weak version; see [6]). For any natural number
m, m > 2, K2m admits an MTP.
Brualdi–Hollingsworth Conjecture (see [4]). If m > 2, then in any proper
edge coloring of K2m with 2m− 1 colors, all edges can be partitioned into m
multi-colored spanning trees.
In [4] it was proved that in any proper edge coloring of K2m(m > 2) with 2m− 1
colors there are at least two edge disjoint multicolored spanning trees.
Constantine’s Conjecture (strong version; see [6]). If m > 2, then in any
proper edge coloring of K2m with 2m− 1 colors, all edges can be partitioned into m
isomorphic multicolored spanning trees.
The main goal of this paper is to prove the first conjecture.
Example 1. The complete graph K6 admits an MTP. To see this consider the
complete graph K6with the vertex set{1, . . . , 6}. Table 2 gives a proper edge coloring
of K6 with colors c1, . . . , c5 as well as an MTP for it. The ith row of this table is
the set of all edges with color ci. Each column denotes the edges of a multicolored
spanning tree. Figure 1 shows that the spanning trees T1, T2, T3 are isomorphic.
In [6] it has been shown that K8admits an MTP.
Using the software Gap, Peter Cameron found a decomposition of K6,6 into six
isomorphic multicolored graphs K1,3∪3K2∪2K1. In the next lemma, using Cameron’s
decomposition we find an MTP for K12.
Lemma 2. The complete graph K12 admits an MTP.
Proof. Consider the complete graph K12with the vertex set{u1, . . . , u6, v1, . . . , v6}.
Table 3 gives a proper edge coloring of K12with colors c1, . . . , c11as well as an MTP
for it. The ith row of this table is the set of all edges with color ci. Each column
denotes the edges of a multicolored spanning tree. Note that the first six rows of the table determine a decomposition of K6,6 into six multicolored subgraphs isomorphic
to K1,3∪ 3K2∪ 2K1.
Now, we are ready to prove our main result. Theorem. For m= 2, K2m admits an MTP.
Proof. First suppose that m is an odd integer. Consider the complete graph K2m defined on the set A∪ B where A = {a1, . . . , am} and B = {b1, . . . , bm}. For
MULTICOLORED PARALLELISMS OF ISOMORPHIC SPANNING TREES 567 Table 3 T1 T2 T3 T4 T5 T6 c1 u2v5 u1v6 u6v1 u3v2 u4v3 u5v4 c2 u2v3 u5v2 u6v6 u4v5 u3v4 u1v1 c3 u4v1 u3v3 u6v4 u1v2 u5v5 u2v6 c4 u1v4 u3v5 u5v3 u6v2 u2v1 u4v6 c5 u2v2 u4v4 u1v5 u5v1 u6v3 u3v6 c6 u5v6 u3v1 u4v2 u2v4 u1v3 u6v5 c7 u3u5 u4u6 u1u2 v3v5 v4v6 v1v2 c8 u2u4 u1u5 u3u6 v2v4 v1v5 v3v6 c9 u2u5 u3u4 u1u6 v2v5 v3v4 v1v6 c10 u2u6 u1u3 u4u5 v2v6 v1v3 v4v5 c11 u1u4 u2u3 u5u6 v1v4 v2v3 v5v6
convenience, let G and H be the complete graphs on the sets A and B, respectively. Since m is odd, G has a total coloring π which uses m colors, 1, . . . , m. Now, define an edge-coloring c of K2m as follows:
(a) For each edge ajak∈ E(G), let c(ajak) = π(ajak).
(b) For each edge bjbk∈ E(H), let c(bjbk) = π(ajak).
(c) For each edge aibi, 1≤ i ≤ m, let c(aibi) = π(ai).
(d) For each edge ajbk, j = k, let c(ajbk) = [k− j]m+ m.
Clearly, c is a proper (2m− 1)-edge-coloring of K2m. It is left to decompose K2m
into m multicolored isomorphic spanning trees. First, for each i ∈ {1, . . . , m}, let
Ti be defined on the set A∪ B and E(Ti) = {aia[i+2t]m, bib[i+2t−1]m, bia[i+2t−1]m, a[i+1]mb[i+2t]m| t = 1, 2, . . . ,
m−1
2 } ∪ {aibi}. It is easy to check that each Ti is a
multicolored spanning tree, and all the Ti’s are isomorphic.
Now, if m is not an odd integer, then 2m = 2tm where t ≥ 2 and m is odd.
In the case where m = 1, t must be at least 3. Then it is a direct consequence of Theorem A. Assume m ≥ 3. Thus K2tm admits an MTP by Theorem A except when m= 3 and t = 2. Since this case can be handled by Lemma 2, we conclude the proof.
Acknowledgments. The first two authors are very grateful to professor Peter Cameron for his fruitful discussions, and we appreciate the helpful comments of the referees.
REFERENCES
[1] S. Akbari and A. Alipour, Transversals and multicolored matchings, J. Combin. Des., 12 (2004), pp. 325–332.
[2] I. Anderson, Combinatorial Designs: Construction Methods, Ellis Horwood Limited, Chi-chester, UK, 1990.
[3] R. C. Bose, S. S. Shrikhande, and E. T. Parker, Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture, Canad. J. Math., 12 (1960), pp. 189–203.
[4] R. A. Brualdi and S. Hollingsworth, Multicolored trees in complete graphs, J. Combin. Theory Ser. B, 68 (1996), pp. 310–313.
[5] P. J. Cameron, Parallelisms of Complete Designs, London Math. Soc. Lecture Notes Series 23, Cambridge University Press, Cambridge, UK, 1976.
[6] G. M. Constantine, Multicolored parallelisms of isomorphic spanning trees, Discrete Math. Theor. Comput. Sci., 5 (2002), pp. 121–125.
[7] H. P. Yap, Total Colourings of Graphs, Lecture Notes in Math. 1623, Springer-Verlag, Berlin, 1996.
