We show that for any colouring of the edges of the complete bipartite graphKn,n with 3 colours there are 5 disjoint monochromatic cycles which together cover all but o(n) of the vertices and 18 disjoint monochromatic cycles which together cover all vertices. keywords: monochromatic cycle partition, Ramsey-type problem, complete bipartite graph MSC 2010: 05C69, 05C75, 05C38.

The Gallai–Ramsey number $$gr_{k}(K_{3}: H_{1}, H_{2}, \cdots , H_{k})$$ is the smallest integer n such that every k-edge-colored $$K_{n}$$ contains either a rainbow $$K_3$$ or monochromatic $$H_{i}$$ in color i for some $$i\in [k]$$ . We define star-critical $$gr_{k}^{*}(K_3: as s $$K_{n}-K_{1, n-1-s}$$ When $$H=H_{1}=\cdots =H_{k}$$ we simply denote $$gr_{k}^{*}(K_{3}: by H)$$ determine numbe...

Given a graph G and positive integer k, define the Gallai-Ramsey number to be minimum of vertices n such that any k-edge coloring $$K_n$$ contains either rainbow (all different colored) triangle or monochromatic copy G. Much like Ramsey numbers, numbers have gained reputation as being very difficult compute in general. As yet, still only precious few sharp results are known. In this paper, we o...

We prove that there exists a constant c such that, for any integer ∆, the Ramsey number of a bipartite graph on n vertices with maximum degree ∆ is less than 2n. A probabilistic argument due to Graham, Rödl and Ruciński implies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of Rödl, Ruciński a...

Let k be a fixed positive integer and let H be a graph with at least k + 1 edges. A local (H, k)-coloring of a graph G is a coloring of the edges of G such that edges of no subgraph of G isomorphic to a subgraph of H are colored with more than k colors. In the paper we investigate properties of local (H, k)-colorings. We prove the Ramsey property for such colorings, establish conditions for the...

dual, 88–89 graph, 3, 67, 76, 238 acyclic, 12, 60 adjacency matrix, 24 adjacent, 3 Ahuja, R.K., 145 algebraic colouring theory, 121 flow theory, 128–143 graph theory, ix, 20–25, 28 planarity criteria, 85–86 algorithmic graph theory, 145, 276–277, 281–282 almost, 238, 247–248 Alon, N., 106, 121–122, 249 alternating path, 29 walk, 52 antichain, 40, 41, 42, 252 Appel, K., 121 arboricity, 61, 99, 1...

A subgraph of an edge-coloured graph is rainbow if all of its edges have different colours. For graphs G and H the anti-Ramsey number ar(G,H) is the maximum number of colours in an edge-colouring of G with no rainbow copy of H. The notion was introduced by Erdős, Simonovits and V. Sós and studied in case G = Kn. Afterwards exact values or bounds for anti-Ramsey numbers ar(Kn, H) were establishe...