Monochromatic triangles in three-coloured graphs

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...

Partitioning two-coloured complete multipartite graphs into monochromatic paths

We show that any complete k-partite graph G on n vertices, with k ≥ 3, whose edges are two-coloured, can be covered by two vertex-disjoint monochromatic paths of distinct colours, under the necessary assumption that the largest partition class of G contains at most n/2 vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, ...

Partitioning 3-coloured complete graphs into three monochromatic paths

Edge colorings of complete graphs without tricolored triangles

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur. 2004 Wiley Periodicals, Inc. J Graph Theory 46: 211–216, 2004

On Tensor Product of Graphs, Girth and Triangles

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.