On monochromatic configurations for finite colorings

Chromatic Sums for Colorings Avoiding Monochromatic Subgraphs

Given graphs G and H, a vertex coloring c : V (G) → N is an H-free coloring of G if no color class contains a subgraph isomorphic to H. The H-free chromatic number of G, χ(H,G), is the minimum number of colors in an H-free coloring of G. The H-free chromatic sum of G, Σ(H,G), is the minimum value achieved by summing the vertex colors of each H-free coloring of G. We provide a general bound for ...

Monochromatic progressions in random colorings

Let N(k) = 2k/2k3/2f(k) and N(k) = 2k/2k1/2 g(k) where f(k) → ∞ and g(k) → 0 arbitrarily slowly as k → ∞. We show that the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic kterm arithmetic progression approaches 0, as k → ...

Vertex Colorings without Rainbow or Monochromatic Subgraphs

This paper investigates vertex colorings of graphs such that some rainbow subgraph R and some monochromatic subgraph M are forbidden. Previous work focussed on the case that R = M . Here we consider the more general case, especially the case that M = K2.

Planar graph colorings without short monochromatic cycles

Monochromatic Cycle Partitions in Local Edge Colorings

An edge colouring of a graph is said to be an r-local colouring if the edges incident to any vertex are coloured with at most r colours. Generalising a result of Bessy–Thomassé and others, we prove that the vertex set of any 2-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number r, we show that the verte...