Monochromatic trees in random graphs

Random Trees in Random Graphs

We show that a random labeled n-vertex graph almost surely contains isomorphic copies of almost all labeled n-vertex trees, in two senses. In the first sense, the probability of each edge occurring in the graph diminishes as n increases, and the set of trees referred to as "almost all" depends on the random graph. In the other sense, the probability of an edge occurring is fixed, and the releva...

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number 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. We determine t2(K(n1, n2, . . . , nk)) of the complete k-partite graph K(n1, n2, . . . , nk). In particular, we prove that t2(K(n,m)) = (m − 2)/2n + 2, whe...

Trees in random graphs

Let us consider the probability space Ifi(n, p) consisting of all graphs on n labeled vertices where each edge occurs with probability p =1q, independently of all other edges . The aim of this note is to find such natural numbers which are likely to occur as orders of maximal induced trees contained in a graph G E 19(n, p) when 0 < p < 1 is fixed . By a maximal induced tree we mean an induced t...

Partitioning Random Graphs into Monochromatic Components

Erdős, Gyárfás, and Pyber (1991) conjectured that every r-colored complete graph can be partitioned into at most r − 1 monochromatic components; this is a strengthening of a conjecture of Lovász (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into r monochromatic components is p...

Multicolored Trees in Random Graphs