Lower Bounds for the Partial Grundy Number of the Lexicographic Product of Graphs

A Grundy coloring of a graph $G$ is a coloring obtained by applying the greedy algorithm according to some order of the vertices of $G$. The Grundy number of $G$ is then the largest $k$ such that $G$ has a greedy coloring with $k$ colors. A partial Grundy coloring is a coloring where each color class contains at least one greedily colored vertex, and the partial Grundy number of $G$ is the largest $k$ for which $G$ has a partial greedy coloring. In this article, we give some results on the partial Grundy number of the lexicographic product of graphs, drawing a parallel with known results for the Grundy number.

Grundy Number of Some Chordal Graphs

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c : V → {1, 2, . . .} such that c(u) 12≠"> c(v) for every edge uv ∈ E. For proper coloring, colors assigned must be minimum, but for Grundy coloring which should be maximum. In this instance, Grundy numbers of chordal graphs like Cartesian product of two path graphs, join of the path and complete graphs and the line graph of tadpole have been executed