Anti-van der Waerden Numbers on Graphs

Abstract

Given a graph G, an exact r-coloring of G is a surjective functionAn arithmetic progression is rainbow if all of the vertices are colored distinctly. The fewest number of colors that guarantees a rainbow arithmetic progression of length three is called the anti-van der Waerden number of G and is denoted aw(G, 3). It is known that 3 ≤ aw(G H, 3) ≤ 4. Here we determine exact values aw(T T ′ , 3) for some trees T and T ′ , determine aw(G T, 3) for some trees T , and determine aw(G H, 3) for some graphs G and H.

Anti-Ramsey problems on graphs and hypergraphs

“…The following conjecture states that the anti-van der Waerden number for trees is closely related to the anti-van der Waerden number for paths. In [7], related results were also shown for binary trees, Cartesian products of graphs, hypercubes, and graphs with dominating vertices.…”

“…The radius of G is the minimum eccentricity of any vertex, and the diameter of G is the maximum eccentricity of any vertex. In [7], the authors proved the following two results relating the anti-van der Waerden number of a graph to radius and diameter. Trees are the main graph family considered in [7].…”

“…In [7], the authors proved the following two results relating the anti-van der Waerden number of a graph to radius and diameter. Trees are the main graph family considered in [7]. A bijacent vertex is a degree two vertex with two degree two neighbors.…”

“…This work is part of the paper "Anti-van der Waerden numbers on graphs," by Zhanar Berikkyzy, Alex Schulte, Elizabeth Sprangel, Shanise Walker, Nathan Warnberg and Michael Young [7].…”

“…[7] If G is a graph with diam(G) = d and G contains at least one k-AP, then k ≤ aw(G, k) ≤ r d (P k ).…”