Partitioning 2-edge-colored graphs by monochromatic paths and cycles

Abstract

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2\alpha(G)$ monochromatic cycles, where $\alpha(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1+\eps){3|V(G)|\over 4}$ can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that $\overline{G}$ does not contain a fixed bipartite graph $H$, we show that in every $2$-edge-coloring of $G$, $|V(G)|-c(H)$ vertices can be covered by two vertex disjoint paths of different colors, where $c(H)$ is a constant depending only on $H$. In particular, we prove that $c(C_4)=1$, which is best possible

Shortcutting directed and undirected networks with a degree constraint

“…Let H p,q,r be the directed tri-partite graph with node partition A ∪ B ∪ C such that (i) |A| = p, |B| = q, and |C| = r, (ii) all nodes of A are linked by an arc to all nodes of B, and (iii) all nodes of B linked by an arc to all nodes of C . The following lemma can be seen as extending Proposition 4.7 in [3].…”

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

“…(Balogh, Barát, Gerbner, Gyárfás, Sárközy [3], 2012) For every η > 0 there is n 0 (η) such that the following holds. For every n-vertex graph G with n ≥ n 0 and δ(G) > ( 3 4 + η)n, in every 2-coloring of G there exists a red and a vertex disjoint blue cycle, covering all but at most ηn vertices of G.…”

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

“…Suppose that 3n − 1 = 4m for some m and consider a graph whose vertex set is partitioned into four parts A 1 vertices, much smaller than 2n, while the minimum degree is 3m − 1 = 3(3n−1) 4 − 1 and the sum of degrees of nonadjacent pairs is 6m − 2 = 3(3n−1) 2 − 2. Thus, a small increase in the minimum degree (or in the sum of degrees of nonadjacent pairs) results a dramatic increase of the length of the longest monochromatic path.…”

“…The material of this section is again fairly standard by now (see e.g. [1,11,12,13,14,15]) so we omit some of the details. The discussion closely follows the treatment in [2] where also an Ore-type condition was transferred to the reduced graph.…”