Monochromatic Tree Partition for Complete. Multipartite Graphs

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 43, Monochromatic Tree Partition for Complete Multipartite Graphs Shili Wen and Peipei Zhu Department of Mathematics Zhejiang Normal University Jinhua , P.R. China Abstract An r-edge-coloring of a graph G is a surjective assignment of r colors to the edges of G. The monochromatic tree partition number of an r- edge-colored graph G is defined to be the minimum positive integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In this paper, we give a direct proof for the monochromatic tree partition number of an r-edge-colored complete multipartite graph K n1,n 2,,n k. Mathematics Subject Classification: 05C35 Keywords: Monochromatic tree, Partition number, Complete multipartite graph

2 2130 Shili Wen and Peipei Zhu 1 Introduction An edge colored graph is called monochromatic if all the edges of it have same colors. An edge colored graph of at most one edge is also regarded as monochromatic. From Erdős s remark, every 2-edge-coloring of K n contains a monochromatic spanning tree. As one of variants of Erdős s remark, an natural problem is to partition an r-edge-colored K n into as few as possible vertex disjoint monochromatic trees. The (monochromatic) tree (path, cycle) partition number of an r-edge-colored graph G is defined to be the minimum positive integer p such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most p vertex disjoint monochromatic trees (path, cycle). Erdős et al. [2] introduced these parameters and proved that the (monochromatic) tree (cycle) partition number of K n is at most cr 2 ln r for some constant c. Erdős et al.[2] also conjectured that the monochromatic cycle partition number of K n is r and the monochromatic tree partition number is r 1. Almost solving one of the two conjectures, Haxell and Kohayakawa [7] proved that the monochromatic tree partition number of K n is at most r provided that n is large enough with respect to r. Haxell [6] proved that the monochromatic cycle partition number of the complete bipartite graph K n,n is also independent of n, which answered a question in [2]. Kaneko et al.[9] gave an explicit expression for the monochromatic tree partition number of a 2-edge-colored complete multipartite graph. The algorithmic aspects of the problem was considered in [8, 10]. Other related partition problems can be found in [1, 3, 11, 13].

3 Monochromatic tree partition Main result Theorem 2.1 Let n 1 n 2 n k, n = n 1 +n 2 + +n k 1, and r 3. If K n1,n 2,,n k is r-edge-colored such that for each color c, there is a vertex u all the edges incident with which are colored with the color c, then the monochromatic tree partition number of K n1,n 2,,n k is (n k r)(r 1) n + r. Proof. Let φ be an r-edge-coloring of the graph K n1,n 2,,n k, where there is a vertex x i, i =1, 2,,r, such that all the edges incident with the vertex x i has the color i, i =1, 2,,r. Clearly, all these vertices x 1,x 2,,x r lie in the same part, say N s. Now we show that under the edge coloring φ, each optimal monochromatic tree partition contains at most (n k r)(r 1) n + r vertex-disjoint monochromatic trees. Under the r-edge-coloring φ, it is easy to see that every optimal monochromatic tree partition has r different color trees which contains the vertices x 1,x 2,,x r, respectively, and the other monochromatic trees consist of the vertices of N s. Now we only need to estimate the number of the monochromatic trees in the optimal monochromatic tree partition, which does not contain any x i, i =1, 2,,r. Let Q = V (K n1,n 2,,n k ) \N s and Q = p. Note that for any optimal monochromatic tree partition P under the edge coloring φ, there exists an ordered r-partition of Q, says 1,S 2,,S r, such that both S i and x i lie in a same tree of P and each tree not containing any x i consists of only one vertex. On the other hand, for each ordered r-partition of Q, says 1,S 2,,S r, we can easily construct a minimal monochromatic tree partition Q with as few trees as possible, such that both S i and x i lie in a same tree of Q and each tree of Q not containing x i consists of only one vertex of N s \{x 1,x 2,,x r }. Clearly, for any y N s \{x 1,x 2,,x r }, the vertex y is a tree in the monochromatic tree partition Q if and only if for any x S i, the edge xy is not colored with the color i.

4 2132 Shili Wen and Peipei Zhu Clearly, there are r p ordered r-partitions of the vertices set Q. For each y N s \{x 1,x 2,,x r }, there are just (r 1) p monochromatic tree partition where y is a tree. In all these minimal monochromatic tree partitions constructed from ordered r-partitions of Q, the total number of trees consisting of a vertex of N s \{x 1,x 2,,x r } is (n s r)(r 1) p. So there exists a monochromatic tree partition R constructed from an ordered r-partition of Q which has at most (ns r)(r 1)p trees consisting of only a vertex of N r p s \{x 1,x 2,,x r }. Therefore, there exists at most (ns r)(r 1)p + r (n k r)(r 1) n + r vertexdisjoint monochromatic trees in each optimal monochromatic tree partition r p of φ. Next, we present an r-edge-coloring where each optimal monochromatic tree partition contains at least (n k r)(r 1) n +r vertex-disjoint monochromatic trees. Let x 1,x 2,,x r N k. Color all the edges incident with x i with the color i, i =1, 2,,r. Color the edges of K n1,n 2,,n k N k arbitrary. Let {y 1,y 2,,y t } = N k r x i. There are ordered r-partitions of the i=1 set V (K n1,n 2,,n k )\N k, denoted by π 1,π 2,,π. Let S i1,s i2,,s ir denoted the ordered r-partition π i. Now we color all the edges incident with the vertices y 1.y 2,,y t, as follows: If j = i mod( ), then color all the edges between y j and S il with the color l, l =1, 2,,r. Denote by ψ the edge coloring obtained above. Let A and B be two minimal monochromatic tree partitions constructed from two different ordered r-partitions of V (K n1,n 2,,n k )\N k. From the definition of the edge coloring ψ, the difference between the numbers of trees in A and B is at most one. It is easy to verify that under the edge coloring ψ, each optimal monochromatic tree partition contains at least (n k r)(r 1) n + r vertex-disjoint monochromatic tree. This completes the proof.

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