The majority coloring of the join and Cartesian product of some digraph

A majority coloring of a digraph is a vertex coloring such that for every vertex, the number of vertices with the same color in the outneighborhood does not exceed half of its out-degree. Kreutzer, Oum, Seymour and van der Zyper proved that every digraph is majority 4colorable and conjecture that every digraph has a majority 3-coloring. This paper mainly studies the majority coloring of the joint and Cartesian product of some special digraphs and proved the conjecture is true for the join graph and the Cartesian product. According to the influence of the number of vertices in digraph, we prove the majority coloring of the joint and Cartesian product of some digraph.


Introduction
Wood [1], who showed that every digraph is majority 4-colorable. They proved this result on the basis that every acyclic digraph is majority 2-colorable. A majority coloring of odd directed cycle must be a proper vertex coloring, three colors are necessary. Therefore, they proposed the following conjecture: Although this conjecture has not been fully resolved, S. Kreutzer et al. [1] studied some special digraphs. The conjecture is true for the digraphs with certain restrictions on outdegree or in-degree. M. Anastos, A. Lamaison, R. Steiner and T. Szabo [2] showed the following theorem: We call a digraph k -majority choosable, if for any assignment of lists of sizes at least k to the vertices, we can choose colors from the respective lists such that the arising coloring is a majority coloring. Anholcer, Bosek and Grytczuk [3] gave a beautiful proof to show that every digraph is majority 4-choosable(not only majority 4-colorable). M. Anastos, A. Lamaison, R. Steiner and T. Szabo [2] showed the following theorem:

Theorem 1.4[2]
Let D be a digraph whose underlying undirected graph is 6-choosable. Then D is majority 3-choosable. In particular any digraph with a 5-degenerate underlying graph is majority 3-choosable.

Theorem 1.5[2]
If D is a digraph without odd directed cycle, then D is majority 2-

choosable.
A. Girão et al. [4] and F. Knox It is obvious that even directed cycles and directed paths are majority 2-colorable, and odd directed cycles are majority 3-colorable. In this paper, We study the majority coloring of the join and Cartesian product of some digraphs. In Section 2, we prove several results about the majority coloring of the join of some digraphs. In Section 3, we prove the result about the majority coloring of the Cartesian product of some digraphs.
.We denote the join of Let 1 2 , D D be digraphs, we can denote the join of 1 2 ,  is a directed cycle with n vertices. Next, we study the majority coloring of the join of directed paths and directed cycles.
If n is even, . Therefore we color m u with color 1. According to this, we alternate the coloring of ( ) Frist, we consider the case if n is even, n C is an even directed cycle, so it is majority 2-colorable, and satisfied If n is odd, n C is an odd cycle, and the out-neighborhood of ( )  Proof. Let We know that n P is majority 2colorable. If n is even,  Next, we consider that n is even, m is odd, n C is an even directed cycle, and the proper 2-coloring of n C must be a majority 2-coloring. We suppose that V

The majority coloring of the Cartesian product of some digraphs
The Cartesian product of graph G and H is a graph that vertex set is We denoted the Cartesian product of graph G and H by G H × .
This definition is extended to digraphs: Let Let , m n∈ Ν , and 2, 3 m n ≥ ≥ , m P is a directed path with m vertices, n C is a directed cycle with n vertices.
Next, we give the conclusion of the majority coloring of the Cartesian product composed of directed path and directed cycle. Proof. It is obviously that The proper 2-coloring of m n P P × must be a majority 2-coloring, hence m n P P × is majority 2-colorable. The theorem is proved. We know that the Cartesian Product satisfies the commutative law, hence the majority coloring of m n P C × and n m C P × is the same. Therefore we only need to discuss one case of two cases, and suppose that we discuss m n P C × .This prove the claim.  When at least one of m and n is even, first we consider that exactly one is even. We can suppose that m is odd, n is even. Every [ ] i D V is an even directed cycle, and every j D W     is an odd directed cycle. The number of vertices with the same color of ( )