Rainbow matchings in bipartite multigraphs

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$'s. Several choices of parameters relate to known results and conjectures

Large monochromatic components in edge colored graphs with a minimum degree condition

It is well-known that in every k-coloring of the edges of the complete graph Kn there is a monochromatic connected component of order at least (formula presented)k-1. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For k = 2 the authors proved that δ(G) ≥(formula presented) ensures a monochromatic connected component with at least δ(G) + 1 vertices in every 2-coloring of the edges of a graph G with n vertices. This result is sharp, thus for k = 2 we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of k the situation is different, graphs of minimum degree (1 − ϵk)n can replace complete graphs and still there is a monochromatic connected component of order at least (formula presented), in fact (formula presented) suffices. Our second result is an improvement of this bound for k = 3. If the edges of G with δ(G) ≥ (formula presented) are 3-colored, then there is a monochromatic component of order at least n/2. We conjecture that this can be improved to 9 and for general k we (onjectu) the following: if k ≥ 3 and G is a graph of order n such that δ(G) ≥ (formula presented) n, then in any k-coloring of the edges of G there is a monochromatic connected component of order at least (formula presented). © 2017, Australian National University. All rights reserved

On reducible and primitive subsets of F_p, II

In Part I of this paper we introduced and studied the notion of reducibility and primitivity of subsets of F_p: a set A is said to be reducible if it can be represented in the form A = B + C with |B|, |C| > 1. Here we introduce and study strong form of primitivity and reducibility: a set A is said to be k-primitive if changing at most k elements of it we always get a primitive set, and it is said to be k - reducible if it has a representation in the form A = B_1 + B_2 + ... + B_k with |B_1|, |B_2|, ..., |B_k| > 1

LARGE FAMILIES OF PSEUDORANDOM SUBSETS FORMED BY POWER RESIDUES

International audienceIn an earlier paper the authors introduced the measures of pseudo-randomness of subsets of the set of the positive integers not exceeding N , and they also presented two examples for subsets possessing strong pseudorandom properties. One of these examples included permutation polynomials f (X) ∈ F p [X] and d-powers in F p. This construction is not of much practical use since very little is known on permutation polynomials and there are only very few of them. Here the construction is extended to a large class of polynomials which can be constructed easily, and it is shown that all the subsets belonging to the large family of subsets obtained in this way possess strong pseudorandom properties. The complexity of this large family is also studied

Partitioning 3-colored complete graphs into three monochromatic cycles

We show in this paper that in every 3-coloring of the edges of Kn all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic cycles must be different then one can cover ( 3 4 − o(1))n vertices and this is close to best possible

Long rainbow cycles in proper edge-colorings of complete graphs

We show that any properly edge-colored Kn contains a rainbow cycle with at least (4=7 − o(1))n edges. This improves the lower bound of n=2 − 1 proved in [1]