The Adjacent Vertex Sum Distinguishing Coloring of the Join and Cartesian Product of Some Digraphs

1-2-3 Conjecture was researched in 2004, that neighbor-sum-distinguishing chromatic number of G is no more than three. In recent years, the authors devoted themselves to the digraph variant of 1-2-3 Conjecture. Digraph D has a k-arc-coloring, which required that in-arc sum is different from out-arc sum for any arc, which be called adjacent vertex sum distinguishing coloring, also be known as (−, +) variant of 1-2-3 Conjecture. Julien Bensmail, Kasper Lyngsie proved that adjacent vertex sum distinguishing chromatic number is no more than three for every nice digraph. And they give a series of results about complexity. Deciding whether chromatic number less than or equal to three holds is NP-complete for nice digraph D, therefore it is meaningful to study the chromatic number equal to two or equal to three. This paper mainly studies (−, +) variant of 1-2- 3 Conjecture of digraph, and discuss some digraphs that adjacent vertex sum distinguishing chromatic number is equal to 2.