On the Grundy Number of a Graph
نویسندگان
چکیده
The Grundy number of a graph G, denoted by Γ (G), is the largest k such that G has a greedy k-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. Trivially Γ (G) ≤ ∆(G) + 1. In this paper, we show that deciding if Γ (G) ≤ ∆(G) is NP-complete. We then show that deciding if Γ (G) ≥ |V (G)| − k is fixed parameter tractable with respect to the parameter k.
منابع مشابه
A linear algorithm for the grundy number of a tree
A coloring of a graph G = (V ,E) is a partition {V1, V2, . . . , Vk} of V into independent sets or color classes. A vertex v ∈ Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj for every j <i. A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy number of a graph is the maximum number of colors in a Grundy c...
متن کاملNew Bounds on the Grundy Number of Products of Graphs
The Grundy number of a graph G is the largest k such that G has a greedy k-coloring, that is, a coloring with k colors obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this article, we give new bounds on the Grundy number of the product of two graphs. 2011 Wiley Periodicals, Inc. J Graph Theory
متن کاملGrundy number of graphs
The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, 1 ≤ i ≤ k, is adjacent to (i− 1) vertices colored with each color j, 1 ≤ j ≤ i− 1. In this paper we give bounds for the Grundy number of some graphs and cartesian products of graphs. In particular, we determine an exact value ...
متن کاملA lower bound for approximating the grundy number
In first-fit coloring, we color the vertices with positive integers. The vertices arrive on-line, and we color the next vertex v by the minimum number not appearing in its neighborhood. The grundy number of a graph is the maximum possible number of colors used by on-line first-fit coloring, under the worst order of arrival of vertices. The grundy number problem is to find this worst ordering. T...
متن کاملComplexity of Grundy Coloring and Its Variants
The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Col...
متن کاملGrundy chromatic number of the complement of bipartite graphs
A Grundy k-coloring of a graph G, is a vertex k-coloring of G such that for each two colors i and j with i < j, every vertex of G colored by j has a neighbor with color i. The Grundy chromatic number Γ(G), is the largest integer k for which there exists a Grundy k-coloring for G. In this note we first give an interpretation of Γ(G) in terms of the total graph of G, when G is the complement of a...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010