Abstract
The All‐Ones Problem comes from the theory of σ+‐automata, which is related to graph dynamical systems as well as the Odd Set Problem in linear decoding. In this paper, we further study and compute the solutions to the “All‐Colors Problem,” a generalization of “All‐Ones Problem,” on some interesting classes of graphs which can be divided into two subproblems: Strong‐All‐Colors Problem and Weak‐All‐Colors Problem, respectively. We also introduce a new kind of All‐Colors Problem, k‐Random Weak‐All‐Colors Problem, which is relevant to both combinatorial number theory and cellular automata theory.
Highlights
A graph dynamical system (GDS) is a dynamical system constructed over a graph whose vertices can have different states, such that all these states together at a given time constitute a state of the system which can evolve according to an updating scheme [1, 2]
We further study the “All-Colors Problem,” a natural generalization of “All-Ones Problem,” which can be divided into two subproblems: Strong-AllColors Problem and Weak-All-Colors Problem, respectively
Note is called that we always assume that n ≥ m and k ≤ m when we discuss on the k-Random WACP on Zm
Summary
A graph dynamical system (GDS) is a dynamical system constructed over a graph whose vertices can have different states, such that all these states together at a given time constitute a state of the system which can evolve according to an updating scheme [1, 2]. Sutner [17] proved that it is always possible to light every lamp in any graphs by σ+ rule. In graph-theoretic terminology, a solution to the AllOnes Problem with σ+-rule can be stated as follows: given a graph G = (V, E), where V and E denote the vertex-set and the edge-set of G, respectively, a subset X of V is a solution if and only if for every vertex V of G the number of vertices in. The All-Ones Problem can be formulated as follows: given a graph G = (V, E), does a subset X of V exist such that, for any vertex V ∈ V−X, the number of vertices in X adjacent to V is odd, while for any vertex V ∈ X, the number of vertices in X adjacent to V is even?.
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