Monochromatic triangles in three-coloured graphs

Monochromatic triangles in three-coloured graphs

In 1959, Goodman [9] determined the minimum number of monochromatic triangles in a complete graph whose edge set is 2-coloured. Goodman [10] also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, for n sufficiently large, we determine the minimum number of monochromatic triangles in a 3-coloured copy of K...

Partitioning 3-coloured complete graphs into three monochromatic paths

Monochromatic Cycles in 2-Coloured Graphs

Li, Nikiforov and Schelp [12] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length `, for all ` ∈ [4, dn/2e]. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with δ(G) = 3n/4 that do not contain all such cycles. Finally, we show that, for all δ > 0 and n > n0(δ), if G is a 2-e...

Edge disjoint monochromatic triangles in 2-colored graphs

Let N (n; k) be the minimum number of pairwise edge disjoint monochromatic complete graphs Kk in any 2-coloring of the edges of a Kn. Upper and lower bounds on N (n; k) will be given for k¿ 3. For k =3, exact values will be given for n6 11, and these will be used to give a lower bound for N (n; 3). c © 2001 Elsevier Science B.V. All rights reserved.

Monochromatic Structures in Edge-coloured Graphs and Hypergraphs - A survey

Given a graph whose edges are coloured, on how many vertices can we find a monochromatic subgraph of a certain type, such as a connected subgraph, or a cycle, or some type of tree? Also, how many such monochromatic subgraphs do we need so that their vertex sets form either a partition or a covering of the vertices of the original graph? What happens for the analogous situations for hypergraphs?...