Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

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

Partitioning a 2-edge-coloured graph of minimum degree 2n/3+o(n)2n/3 + o(n) into three monochromatic cycles

Lehel conjectured in the 1970s that every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomass\'e. However, the host graph $G$ does not have to be complete. It it suffices to require that $G$ has minimum degree at least $3n/4$, where $n$ is the order of $G$, as was shown recently by Letzter, confirming a conjecture of Balogh, Bar\'{a}t, Gerbner, Gy\'arf\'as and S\'ark\"ozy. This degree condition is asymptotically tight. Here we continue this line of research, by proving that for every red and blue edge-colouring of an $n$-vertex graph of minimum degree at least $2n/3 + o(n)$, there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight

Partitioning a graph into a cycle and a sparse graph

In this paper we investigate results of the form “every graph G has a cycle C such that the induced subgraph of G on V (G) \ V (C) has small maximum degree.” Such results haven’t been studied before, but are motivated by the Bessy and Thomassé Theorem which states that the vertices of any graph G can be covered by a cycle C1 in G and disjoint cycle C2 in the complement of G. There are two main theorems in this paper. The first is that every graph has a cycle with (G[V (G) \ V (C)]) ≤ 1 2 (|V (G) \ V (C)| − 1). The bound on the maximum degree (G[V (G) \ V (C)]) is best possible. The second theorem is that every k-connected graph G has a cycle with (G[V (G) \ V (C)]) ≤ 1 k+1 |V (G) \ V (C)| + 3. We also give an application of this second theorem to a conjecture about partitioning edge-coloured complete graphs into monochromatic cycle

Ramsey number of paths and connected matchings in Ore-type host graphs

It is well-known (as a special case of the path-path Ramsey number) that in every 2-coloring of the edges of K3(n-1), the complete graph on 3n - 1 vertices, there is a monochromatic P-2n, a path on 2n vertices. Schelp conjectured that this statement remains true if K3n-1 is replaced by any host graph on 3n - 1 vertices with minimum degree at least 3(3n-1)/4. Here we propose the following stronger conjecture, allowing host graphs with the corresponding Ore-type condition: If G is a graph on 3n - 1 vertices such that for any two non-adjacent vertices u and v, d(G)(u) + d(G)(v) >= 3/2 (3n - 1), then in any 2-coloring of the edges of G there is a monochromatic path on 2n vertices. Our main result proves the conjecture in a weaker form, replacing P-2n by a connected matching of size n. Here a monochromatic, say red, matching in a 2-coloring of the edges of a graph is connected if its edges are all in the same connected component of the graph defined by the red edges. Applying the standard technique of converting connected matchings to paths with the Regularity Lemma, we use this result to get an asymptotic version of our conjecture for paths. (C) 2016 Elsevier B.V. All rights reserved