Signed coloring of 2dimensional grids Abstract   A signed graph is a pair    (  G  ,  xcfx83  )   , where    G    (  V  (  G  )  ,  E  (  G  )  )    is an undirected graph and    xcfx83  :  E  (  G  )  xe2x86x92      ,  xe2x88x92      is a function which marks each edge with xe2x80x9cxe2x80x9d or xe2x80x9cxe2x88x92xe2x80x9d. Two signed graphs are equivalent if one of them can be changed to the other by a sequence of resigning operations. The single resigning operation chooses a vertex    v  xe2x88x88  V  (  G  )    and flips the signs of all edges incident to v. By      G  ,  xcfx83      we shall denote the equivalence class of the signed graph    (  G  ,  xcfx83  )   . Each element of      G  ,  xcfx83      is called a presentation of      G  ,  xcfx83     . In this paper we shall call both    (  G  ,  xcfx83  )    and      G  ,  xcfx83      signed graphs. The coloring of signed graphs is defined through homomorphism. The signed graph      G  ,  xcfx83      is colored by the signed graph    (    G    2    ,    xcfx83    2    )   , if there exists a presentation    (  G  ,    xcfx83    1    )    of      G  ,  xcfx83      and a vertexmapping xcfx95 from G to      G    2      which preserves signs of the edges.  In this paper we show that: (a) The signed chromatic number for the class    G    of all 2dimensional grids lies between 5 and 6. (b) Every signed grid with at most seven rows can be colored with five colors. (c) Every signed grid with two rows can be colored with four colors.

On the Complexity of Digraph Colourings and Vertex Arboricity It has been shown by Bokal et al. that deciding 2colourability of  :12,an NPcomplete problem. This result was later on extended by Feder et al.  that deciding :26,whether a digraph has a circular pcolouring  :36,91,110,131,for all rational pu003e1. In this paper, we consider the  :47,corresponding decision problems :36,91,110,131,for related notions of fractional  :56,digraphs and graphs, including the star dichromatic number, the  :66,number and the circular vertex arboricity. We prove the   :76,Deciding if the star dichromatic :66,number of a digraph is at most p  :36,91,110,131,for every rational   :76,Deciding if the fractional dichromatic :66,number of a digraph is at most p  :36,91,110,131,for every :95,135,pu003e1, p neq   :76,Deciding if the circular vertex :116,arboricity of a graph is at most p  :36,91,110,131,for every rational   To show these results, different techniques are required in each case.  :148,to prove the first result, we relate the star dichromatic :66,number :148,to a  of :162,homomorphisms between digraphs, called circular homomorphisms,  :170,be of independent interest. We provide a classification of  :180,complexities of the :47,corresponding homomorphism colouring  :148,to the :187,one derived by Feder et al. :36,91,110,131,for acyclic homomorphisms.